Smoothly clipped absolute deviation estimation for uncertain autoregressive model

1 Introduction

Relationships between a sequential set of past data measured over time to forecast future values are investigated by time series forecasting. The classical time series forecasting model is autoregressive firstly proposed by Yule [11]. Following that, the improved autoregressive model, such as moving average [10], autoregressive moving average, and autoregressive integrated moving average [7], were investigated. These models have been widely used in various applications, as can be seen in economics, control system, medicine, astronomy, and many others.

In time series modeling applications, choosing the proper order of the model is a fundamental step. It has been of considerable interest by many researchers, such as Akaike [12, 13], Liang et al. [8], Fuchs [17], Sadabadi et al. [19], Al-Smadi and Al-Zaben [1], and Schwarz [9]. The commonly used methods include Akaike’s information criterion (AIC), Akaike’s final prediction error (FPE), and Bayesian information criterion (BIC). Parameter estimation is another important topic and has also received extensive attention in time series analysis. Several methods for estimating parameters of the model have been proposed by Reinsel et al. [6], Paarmann and Korenberg [18], Macdonald [14], Daris [15], and Cochrane and Orcutt [5], but some researchers prefer method of estimation based on least-squares because this method is relatively simple and has an excellent overall performance.

Although the classical time series forecasting methods are widely used and have made great success in the application, they primarily deal with the forecasting problems in which the observed values are real numbers. In practice, the observed data are sometimes incomplete, missing, or censored due to various factors’ influence. Among them, interval-valued data are encountered frequently in multiple situations such as the diastolic and systolic blood pressure in a given day, the variation in an electric current, the company’s daily stock price, and the inventory demand of enterprises in a specific period, etc. It should be noted that such data are human linguistic data that describe the uncertainty or variability of the observed values instead of generic numeric intervals. For this kind of imprecise observed values, describing it and constructing its linguistic model are essential. Therefore, Liu [2] proposed the uncertainty theory based on normality, duality, subadditivity, and product axioms and considered the imprecise observed values as the uncertain variables.

On this basis, Yang and Liu [20] introduced a concept of uncertain time series, proposed an uncertain autoregressive model, and presented the least-squares method for estimating the unknown parameters in the uncertain autoregressive model. Following that, Yang and Ni [21] investigated an uncertain moving average model to deal with uncertain time series problems and presented the least-squares estimation method. Afterward, noting that the least-squares estimation is sensitive to outliers’ presence, the least absolute deviations estimation method was proposed in [22]. However, it isn’t easy to find the optimal solution in the process of calculation. To this end, Chen and Yang [4] proposed the maximum likelihood method to determine the uncertain autoregressive model’s parameters. By this method, the unknown autoregressive parameters and the parameters of uncertainty distributions of the disturbance terms can both be obtained. Furthermore, Zhang et al. [24] presented a least absolute shrinkage and selection operator (LASSO) estimation of the uncertain autoregressive model. For the LASSO obtained by adding the penalty term, the solution typically has many of the coefficients equal to zero, some information about the corresponding observations might be overlooked. To overcome this shortcoming, Chen and Yang [3] gave the ridge method to compute the unknown parameters in the uncertain autoregressive model. Ridge estimation can shrink the large coefficients but does not reach zero, which indicates that there is no feature of variable selection.

To summarize, in the parameter estimation methods mentioned above, the researchers assume that the order of the uncertain autoregressive model is known in advance, and then the parameters were estimated according to this assumption. Practically, the order is unknown, so it has to be correctly found to estimate the model’s correct parameters. To this end, Liu and Yang [23] investigated the cross-validation method to confirm autoregressive model order. Zhang et al. [24] presented the final prediction error criterion to determine the order of the model. Further, this paper proposes a new SCAD estimation method for simultaneously determining the model order and unknown parameters of the uncertain autoregressive model for a pre-given λ, which can overcome the defects of the ordinary LASSO estimator and the ridge estimator. The rest of this paper is organized as follows. In Section 2, the SCAD estimation method to determine the order and unknown parameters in the uncertain autoregressive model are presented. The iterative algorithm is proposed for solving the minimizing penalized estimator. After that, the sum of the squared error approach for selecting an optimal λ is discussed in Section 3. Then the simulation studies and numerical comparisons are given in Section 4. Finally, some conclusions are presented in Section 5.

2 SCAD estimation

This section first reviews the uncertain autoregressive model and presents the SCAD estimation method to determine the order and unknown parameters in the autoregressive model. Then, we give the iterative algorithm for optimizing penalized likelihood functions. Finally, the forecast value and the confidence interval of the future value are given.

An uncertain time series is a sequence of imprecisely observed values that are characterized in terms of uncertain variables, i.e., X = { X 1 , X 2 , ⋯ , X n } where X _t are uncertain variables at times t, t = 1, 2, ⋯ , n, t = 1 , 2 , ⋯ , n , respectively.

Uncertain autoregressive model is commonly used to describe and predict uncertain time series. The actual equation for the uncertain autoregressive model is as follows, X t = φ 0 + φ 1 X t - 1 + ⋯ + φ p X t - p + ɛ t , t = p + 1 , p + 2 , ⋯ , n(1) where φ₀, φ₁, ⋯ , φ _p are unknown parameters, ɛ _t is an uncertain variable, and p is called the order of the uncertain autoregressive model. In uncertain time series analysis, the order of an uncertain autoregressive model is often unknown. This allows the possibility of choosing an under-fitting or over-fitting model, which leads to the reduction of prediction accuracy. Therefore, a key point using the uncertain autoregressive model to fit the data is selecting the order and estimating coefficients simultaneously. To this end, we shall propose a penalized least-squares estimation based on the SCAD penalty that shrinks some regression coefficients to zero and estimates nonzero coefficients to recover the actual model. A form of penalized least-squares is as follows.

Definition 1. Let {X₁, X₂, ⋯ , X _n } be imprecisely observed values characterized in terms of uncertain variables, which satisfy the uncertain autoregressive model (1). Then the SCAD estimation ( φ ˆ 0 , φ ˆ 1 , ⋯ , φ ˆ p ) for (φ₀, φ₁, ⋯ , φ _p ) is defined as follows, ( φ ˆ 0 , φ ˆ 1 , … , φ ˆ p ) = arg min φ 0 , φ 1 , ⋯ , φ p { ∑ t = p + 1 n E [ ( X t - φ 0 - ∑ i = 1 p φ i X t - i ) 2 ] + ( n - p ) ∑ j = 0 p P λ ( | φ j | ) }(2) where P λ ( | φ j | ) = { λ | φ j | , if | φ j | ≤ λ - ( | φ j | 2 - 2 a λ | φ j | + λ 2 ) 2 ( a - 1 ) , if λ < | φ j | ≤ a λ ( a + 1 ) λ 2 2 , if | φ j | > a λ(3) is called the SCAD penalty function, where a > 2 and λ > 0 are two tune parameters.

Theorem 1. Let {X₁, X₂, ⋯ , X _n } be imprecisely observed values characterized in terms of independent uncertain variables with regular uncertain distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively, which satisfy the uncertain autoregressive model (1). Then the SCAD estimation ( φ ˆ 0 , φ ˆ 1 , ⋯ , φ ˆ p ) for (φ₀, φ₁, ⋯ , φ _p ) is the optimal solution of the following problem min φ 0 , φ 1 , ⋯ , φ p { ∑ t = p + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α + ( n - p ) ∑ j = 0 p P λ ( | φ j | ) }(4) where ϒ t - i - 1 ( α , φ i ) = { Φ t - i - 1 ( 1 - α ) , if φ i ≥ 0 Φ t - i - 1 ( α ) , if φ i < 0(5) for i = 1, 2, ⋯ , p .

Proof. For each t, the uncertain variable X t - φ 0 - ∑ i = 1 p φ i X t - i(6) is increasing with respect to X _t and decreasing with respect to X_t-i when φ _i  ≥ 0 or increasing with respect to X_t-i when φ _i  < 0. According to the operational laws of uncertain variable, the inverse uncertainty distributions of the uncertain variable (6) is Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) where ϒ t - i - 1 ( α , φ i ) = { Φ t - i - 1 ( 1 - α ) , if φ i ≥ 0 Φ t - i - 1 ( α ) , if φ i < 0 for i = 1, 2, ⋯ , p and t = p + 1, p + 2, ⋯ , n. By the formula of the second moment, we obtain E [ ( X t - φ 0 - ∑ i = 1 p φ i X t - i ) 2 ] = ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α .(7) Hence the solution of equation (4) is equivalent to that of equation (2). Then the theorem is verified. □

However, finding the estimator of coefficients that minimizes the objective function (4) faces some challenges because the penalty function is not differentiable at the origin. Following Fan and Li [16], we use an iterative algorithm based on a local quadratic approximation to calculate the SCAD estimator for a given λ. Suppose that the initial value φ ˆ j 0 ( j = 0 , 1 , ⋯ , p ) is obtained by the following least-squares estimation ( φ ˆ 00 , φ ˆ 10 , ⋯ , φ ˆ p 0 ) = arg min φ 0 , φ 1 , ⋯ , φ p { ∑ t = p + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α } where ϒ t - i - 1 ( α , φ i ) = { Φ t - i - 1 ( 1 - α ) , if φ i ≥ 0 Φ t - i - 1 ( α ) , if φ i < 0 for i = 1, 2, ⋯ , p and t = p + 1, p + 2, ⋯ , n. If φ ˆ j 0 is very close to 0, then set φ ˆ j = 0 ( j = 0 , 1 , ⋯ , p ) . Otherwise they can be locally approximated by a quadratic function as [ P λ ( | φ j | ) ] ′ = P λ ′ ( | φ j | ) sgn ( φ j ) ≈ { P λ ′ ( | φ j 0 | ) / | φ j 0 | } φ j , φ j ≈ φ j 0 , φ j ≠ 0 where [P _λ  (|φ _j |)] ′ is first derivative of penalty function P _λ  (|φ _j |) and sgn ( φ j ) = { 1 , if φ j > 0 - 1 , if φ j < 0 . By using the Taylor expansion of P _λ  (|φ _j |), we get P λ ( | φ j | ) ≈ P λ ( | φ j 0 | ) + [ P λ ( | φ j | ) ] ′ | φ j = φ j 0 ( φ j - φ j 0 ) + 1 2 [ P λ ( | φ j | ) ] ″ | φ j = φ j 0 ( φ j - φ j 0 ) 2 ≈ P λ ( | φ j 0 | ) + { P λ ′ ( | φ j 0 | ) / | φ j 0 | } φ j 0 ( φ j - φ j 0 ) + 1 2 { P λ ′ ( | φ j 0 | ) / | φ j 0 | } ( φ j - φ j 0 ) 2 = P λ ( | φ j 0 | ) + 1 2 { P λ ′ ( | φ j 0 | ) / | φ j 0 | } ( φ j 2 - φ j 0 2 ) , φ j ≠ 0 . Therefore , the minimization problem ( 4 ) can be reduced to a quadratic minimization problem . Calculating the first and second - order differential of the objective function ( 4 ) , respectively . Setting f ( φ 0 , φ 1 , ⋯ , φ p ) = ∑ t = p + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α + ( n - p ) ∑ j = 0 p P λ ( | φ j | ) . Therefore, its first and second differential can be represented as ∂ f ∂ φ 0 = 2 ∑ t = p + 1 n ∫ 0 1 - ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) d α + ( n - p ) P λ ′ ( | φ 00 | ) | φ 00 | φ 0 , ∂ f ∂ φ i = 2 ∑ t = p + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) · ( - ϒ t - i - 1 ( α , φ i ) ) d α + ( n - p ) P λ ′ ( | φ i 0 | ) | φ i 0 | φ i ,

∂ 2 f ∂ φ 0 2 = ( n - p ) ( 2 + P λ ′ ( | φ 00 | ) | φ 00 | ) ,

∂ 2 f ∂ φ i 2 = 2 ∑ t = p + 1 n ∫ 0 1 ( ϒ t - i - 1 ( α , φ i ) ) 2 d α + ( n - p ) P λ ′ ( | φ i 0 | ) | φ i 0 | .

The corresponding Newton Raphson algorithm is presented below.

Initialize parameters. Setting ( φ ˆ 00 , φ ˆ 10 , ⋯ , φ ˆ p 0 ) = arg min φ 0 , φ 1 , ⋯ , φ p { ∑ t = p + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α } where ϒ t - i - 1 ( α , φ i ) = { Φ t - i - 1 ( 1 - α ) , if φ i ≥ 0 Φ t - i - 1 ( α ) , if φ i < 0 for i = 1, 2, ⋯ , p and t = p + 1, p + 2, ⋯ , n.

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Fig. 1

The iterative algorithm of the proposed method.

Furthermore, suppose that ɛ _t have the same expected value and variance for all t, i.e., E ( ɛ t ) = e , Var ( ɛ t ) = σ 2 , t = p + 1 , p + 2 , ⋯ , n . By the definition of Yang and Liu [20], the expected value of ɛ _t can be estimated as e ˆ = 1 n - p ∑ t = p + 1 n E [ ɛ ˆ t ](8) and the variance can be estimated as σ ˆ 2 = 1 n - p ∑ t = p + 1 n E [ ɛ ˆ t - e ˆ ] 2(9) where ɛ ˆ t is the t-th residual errors that is ɛ ˆ t = X t - φ ˆ 0 - ∑ i = 1 p φ i ˆ X t - i , t = p + 1 , p + 2 , ⋯ , n . In addition, the fitted model is X t = φ ˆ 0 + ∑ i = 1 p φ ˆ i X t - i , t = p + 1 , p + 2 , … n . Then, based on X₁, X₂, ⋯ , X _n , the forecast uncertain variable X ˆ n + 1 of the next uncertain variable X_n+1 can be determined by X ˆ n + 1 = φ ˆ 0 + ∑ i = 1 p φ ˆ i X n - i + ɛ n + 1 and the forecast value μ_n+1 of X_n+1 is defined by μ n + 1 = E [ X ˆ n + 1 ] = φ ˆ 0 + ∑ i = 1 p φ ˆ i E [ X n - i ] + e ˆ .(10) If we assume that the uncertainty distribution of error term ɛ_n+1 is known, then the uncertainty distribution of X ˆ n + 1 can be obtained as Φ ˆ n + 1 ( x ) . According to the Yang and Liu [20], the α-confidence interval of X_n+1 is [ μ n + 1 - b , μ n + 1 + b ](11) where b is the minimum value such that Φ ˆ n + 1 ( μ n + 1 + b ) - Φ ˆ n + 1 ( μ n + 1 - b ) ≥ α .

In this section, we present the sum of the squared error approach to select the optimal λ.

As we can see, before we apply the algorithm mentioned above to compute the SCAD estimator, we need to choose the tuning parameter a and λ. The tuning parameter a = 3.7 suggested in Fan and Li [16] will be used in the whole paper. The tuning parameter λ controls the size of the penalty and the number of selected variables. As a result, choosing the proper value of the tuning parameter λ is an important part of the fitting. In this paper, we use the sum of the squared error method to select the optimal tuning parameter λ in the SCAD estimator. The details of the procedures are as follows. The full dataset n is divided into k subsets of approximately the same size, and training and test set are denoted by n ^v and (n ^v )  ^c , respectively, for v = 1, 2, ⋯ , k. For each λ and v, we obtain the estimator φ ˆ j v of φ _j   (j = 0, 1, ⋯ , n) using the training set n ^v . Then the tuning parameter λ can be selected by minimizing the sum of the squared error SSE (λ), SSE ( λ ) = ∑ v = 1 k ∑ ( n v ) c E [ ( X t - φ ˆ 0 v - ∑ i = 1 p φ ˆ i v X t - i ) 2 ] .(12) According to the equation (7), the SSE (λ) can be calculated as SSE ( λ ) = ∑ v = 1 k ∑ ( n v ) c ∫ 0 1 ( Φ t - 1 ( α ) - φ ˆ 0 v - ∑ i = 1 p φ ˆ i v ϒ t - i - 1 ( α , φ ˆ i v ) ) 2 d α(13) where ϒ t - i - 1 ( α , φ ˆ i v ) = { Φ t - i - 1 ( 1 - α ) , if φ ˆ i v ≥ 0 Φ t - i - 1 ( α ) , if φ ˆ i v < 0 for i = 1, 2, ⋯ , p .

This section will use two numerical examples and a real data example to demonstrate the effectiveness of the proposed method. Furthermore, we make a comparative analysis with other methods.

4.1 Example

Example 1. In this example, a set of imprecisely observed values X₁, X₂, ⋯ , X₂₆ are generated from the following uncertain autoregressive model X t = 15 + 0.09 X t - 1 + 0.84 X t - 2 , t = 3 , 4 , ⋯ , 26 .(14) Here, we suppose that X₁, X₂, ⋯ , X₂₆ are independent linear uncertain variables with linear uncertainty distributions Φ₁, Φ₂, ⋯ , Φ₂₆, respectively, which are listed in Table 1. Let us show how to use the proposed method to determine the model order and estimate the unknown parameters. Assuming that the maximum lag order of the uncertain autoregressive model is p, according to the empirical value, we set p = 5, the corresponding fitted model is X t = φ ˆ 0 + φ ˆ 1 X t - 1 + φ ˆ 2 X t - 2 + φ ˆ 3 X t - 3 + φ ˆ 4 X t - 4 + φ ˆ 5 X t - 5 , t = 6 , 7 , ⋯ , 26 .

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Table 1

Imprecisely observed data where L ( a , b ) represents linear uncertain variable

X 1	X 2	X 3	X 4	X 5	X 6	X 7
L ( 287 , 310 )	L ( 299 , 325 )	L ( 283 , 305 )	L ( 292 , 315 )	L ( 279 , 299 )	L ( 285 , 307 )	L ( 275 , 294 )
X 8	X 9	X 10	X 11	X 12	X 13	X 14
L ( 279 , 299 )	L ( 271 , 289 )	L ( 274 , 292 )	L ( 267 , 284 )	L ( 269 , 286 )	L ( 264 , 279 )	L ( 265 , 281 )
X 15	X 16	X 17	X 18	X 19	X 20	X 21
L ( 260 , 275 )	L ( 261 , 275 )	L ( 257 , 271 )	L ( 257 , 271 )	L ( 254 , 267 )	L ( 254 , 266 )	L ( 251 , 263 )
X 22	X 23	X 24	X 25	X 26
L ( 251 , 262 )	L ( 249 , 260 )	L ( 248 , 259 )	L ( 246 , 256 )	L ( 246 , 255 )

According to the Step 1 in Section 2, the least-squares estimation without penalty function can be obtained as arg min φ 0 , φ 1 , ⋯ , φ 5 { ∑ t = 6 26 ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 5 φ i ϒ t - i - 1 ( α , φ i ) ) 2 d α } where ϒ t - i - 1 ( α , φ i ) = { Φ t - i - 1 ( 1 - α ) , if φ i ≥ 0 Φ t - i - 1 ( α ) , if φ i < 0 for i = 1, 2, ⋯ , 5 and t = 6, 7, ⋯ , 26. Then, we obtain the least-squares estimation as ( φ ˆ 00 , φ ˆ 10 , φ ˆ 20 , φ ˆ 30 , φ ˆ 40 , φ ˆ 50 ) = ( 15.2035 , 0.0929 , 0.8365 , 0.0003 , - 0.0005 , - 0.00001 ) . According to the Newton Raphson algorithm in Section 2, taking the least-squares estimation ( φ ˆ 00 , φ ˆ 10 , φ ˆ 20 , φ ˆ 30 , φ ˆ 40 , φ ˆ 50 ) as the initial value of SCAD estimation. Next, according to the sum of the squared error method, we divide the 26 data sets into 2 subsets of approximately the same size, the first 10 data sets, and the last 3 data sets are used as training and test set, respectively. Therefore, the optimal tuning parameter λ can be selected via minimizing SSE ( λ ) = ∑ v = 1 2 ∑ ( n v ) c ∫ 0 1 ( Φ t - 1 ( α ) - φ 0 - ∑ i = 1 p φ ˆ i v ϒ t - i - 1 ( α , φ ˆ i v ) ) 2 d α where ϒ t - i - 1 ( α , φ ˆ i v ) = { Φ t - i - 1 ( 1 - α ) , if φ ˆ i v ≥ 0 Φ t - i - 1 ( α ) , if φ ˆ i v < 0 for i = 1, 2, ⋯ , 5 . Result for SSE (λ) with λ ∈ {1, 2, ⋯ , 10} is given in Table 2 and Fig. 2. As we can see, taking λ = 4 smallest we get the SSE (λ) among λ ∈ {1, 2, ⋯ , 10} . Therefore, we obtain the corresponding SCAD estimation with λ = 4 as ( φ ˆ 0 , φ ˆ 1 , φ ˆ 2 , φ ˆ 3 , φ ˆ 4 , φ ˆ 5 ) = ( 15.2448 , 0.0924 , 0.8365 , 0 , 0 , 0 ) . Hence the fitted uncertain autoregressive model is X t = 15.2448 + 0.0924 X t - 1 + 0.8365 X t - 2 .

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Table 2

Sum of squared error SSE (λ) for Example 1

SSE (1)	SSE (2)	SSE (3)	SSE (4)	SSE (5)
219.6165	219.3512	219.0548	218.7235	797.4074
SSE (6)	SSE (7)	SSE (8)	SSE (9)	SSE (10)
232.8729	232.8897	232.9095	232.9323	232.9580

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Fig. 2

SSE (λ) with respect to λ for Table 2.

The results show that the SCAD estimation is more close to true parameter value (φ₀, φ₁, φ₂) = (15, 0.09, 0.84) than least-squares estimation, which enables estimate the correct parameters of the model and improves the precision of forecasting. In the following, we further use the SCAD method’s estimated values to calculate the prediction value and prediction interval of the next period. It follows from Equations (8) and (9) that e ˆ = 1 24 ∑ t = 3 26 ∫ 0 1 ( Φ t - 1 ( α ) - φ ˆ 0 - ∑ i = 1 2 φ ˆ i Φ t - i - 1 ( 1 - α ) ) d α = 0.0452 . Therefore, the forecast value of X₂₇ is μ 27 = φ ˆ 0 + φ ˆ 1 E [ X 26 ] + φ ˆ 2 E [ X 25 ] + e ˆ = 248.6793 . It is almost equal to the actual average value 248.6675 obtained by the model (14). For the confidence level α = 0.80, if we further suppose that the disturbance term ɛ₂₇ is a normal uncertain variable with uncertain distribution N ( e ˆ , σ ˆ ) , then we get the prediction interval for X₂₇ is 248.6793 ± 14.9048 .

Example 2.

In this example, we consider a set of historical stock prices in AMZN company. a Here, we use the linear uncertain variables X t ∼ L ( a t , b t ) to describe daily prices over the period from 02 January to 31 December in 2019 and thus n = 252, where a _t and b _t represent the lowest and highest values in the t-th day, t = 1, 2, ⋯ , 252, respectively, which are shown in Fig. 3. To obtain the optimal uncertain autoregressive model for stock price prediction, we assume that the fitted model is X t = φ ˆ 0 + φ ˆ 1 X t - 1 + φ ˆ 2 X t - 2 + φ ˆ 3 X t - 3 + φ ˆ 4 X t - 4 + φ ˆ 5 X t - 5 , t = 6 , 7 , ⋯ , 252 .

For a set of λ, it follows from the equation (13) and the Newton Raphson algorithm that SSE ( λ ) = ∑ v = 1 2 ∑ ( n v ) c ∫ 0 1 ( Φ t - 1 ( α ) - φ ˆ 0 - ∑ i = 1 p φ ˆ i v ϒ t - i - 1 ( α , φ ˆ i v ) ) 2 d α where ϒ t - i - 1 ( α , φ ˆ i v ) = { Φ t - i - 1 ( 1 - α ) , if φ ˆ i v ≥ 0 Φ t - i - 1 ( α ) , if φ ˆ i v < 0 for i = 1, 2, ⋯ , 5 . Result for SSE (λ) with λ ∈ {1, 2, ⋯ , 10} is given in Table 3 and Fig. 4.

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Table 3

Sum of squared error SSE (λ) for Example 2

SSE (1)	SSE (2)	SSE (3)	SSE (4)	SSE (5)
4702.4820	4655.3256	4656.9152	4659.3069	4661.3951
SSE (6)	SSE (7)	SSE (8)	SSE (9)	SSE (10)
4664.5000	4667.3336	4670.1673	4673.7347	13134.9381

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Fig. 3

Part of the interval-valued time series of the AMZN stock prices.

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Fig. 4

SSE (λ) with respect to λ for Table 3.

Therefore, the tuning parameter λ = 2 is chosen for SCAD estimation and the fitted model as X t = 29.2634 + 0.2342 X t - 1 + 0.5898 X t - 2 + 0.1454 X t - 3 , t = 4 , 5 , ⋯ , 252 . By using Equations (8) and (9), e ˆ = 1 249 ∑ t = 4 252 ∫ 0 1 ( Φ t - 1 ( α ) - ∑ i = 1 3 φ ˆ i Φ t - i - 1 ( 1 - α ) ) d α = 27.8947 and σ ˆ 2 = 1 249 ∑ t = 4 252 ∫ 0 1 ( Φ t - 1 ( α ) - ∑ i = 1 3 φ ˆ i Φ t - i - 1 ( 1 - α ) - e ˆ ) 2 d α = 1191.2 . Therefore, the forecast value of X₂₅₃, i.e., the stock price on January 2, 2020 is μ 253 = φ ˆ 0 + φ ˆ 1 E [ X 252 ] + φ ˆ 2 E [ X 251 ] + φ ˆ 3 E [ X 250 ] + e ˆ = 1861.0101 . For the confidence level α = 0.80, if we further suppose that the disturbance term ɛ₂₅₃ is a normal uncertain variable with uncertain distribution N ( e ˆ , σ ˆ ) , then we get the prediction interval for X₂₅₃ is 1861.0101 ± 56.0730 . To further assess the proposed method’s effectiveness, we calculate the stock price forecasts for January 2 to January 30, 2020, by using the least square method and SCAD method, respectively, and the results are shown in Table 4. As we can see, the SCAD estimation method’s performance is better than the least-squares estimation method. The possible reason is that the SCAD estimation method can shrink some nonsignificant coefficients to zero and retain the nonzero coefficients to establish a truly effective model and improve prediction accuracy.

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Table 4

Comparison of SCAD estimation and least-squares estimation method

Time	Actual average values**	SCAD estimation method		Least-squares estimation method
		Predicted values	Absolute deviation	Predicted values	Absolute deviation
1/2/2020	1881.08	1861.01	20.07	1860.39	20.70
1/3/2020	1875.35	1850.64	24.71	1849.87	25.48
1/6/2020	1881.85	1856.14	25.71	1854.96	26.88
1/7/2020	1902.97	1853.96	49.00	1852.52	50.45
1/8/2020	1898.72	1855.19	43.53	1853.42	45.30
1/9/2020	1906.81	1854.99	51.82	1852.93	53.88
1/10/2020	1893.47	1855.35	38.12	1852.99	40.48
1/13/2020	1889.4	1855.50	33.90	1852.85	36.55
1/14/2020	1872.83	1855.72	17.11	1852.78	20.05
1/15/2020	1866.98	1855.91	11.06	1852.69	14.29
1/16/2020	1875.81	1856.11	19.70	1852.61	23.20
1/17/2020	1871.95	1856.30	15.65	1852.52	19.42
1/21/2020	1877.14	1856.48	20.65	1852.44	24.69
1/22/2020	1892.92	1856.67	36.25	1852.36	40.56
1/23/2020	1881.37	1856.85	24.52	1852.28	29.08
1/24/2020	1871.22	1857.03	14.19	1852.21	19.01
1/27/2020	1828.17	1857.20	29.03	1852.13	23.96
1/28/2020	1844.07	1857.38	13.31	1852.06	7.99
1/29/2020	1864.89	1857.55	7.34	1851.98	12.90
1/30/2020	1861.74	1857.71	4.03	1851.91	9.83
Total absolute deviation			499.70		544.70

^**Average value of the highest and lowest price of the stock.

In this paper, we presented the SCAD estimation method to determine the uncertain autoregressive model’s parameters. This method can get both the order and the unknown coefficients of the uncertain autoregressive model. Furthermore, we presented the sum of the squared error method to find the optimal tuning parameter λ. Finally, a simulation study and a real data example were conducted to verify the effectiveness of the proposed method, and a comparative analysis is analyzed to illustrate the advantage of our approach in the presence of selecting the order and estimating parameters in an uncertain autoregressive model.