Uncertain max-autoregressive model with imprecise observations

1 Introduction

Real decisions are usually made in the uncertain state (e.g., war, flood, and even rumor), which makes raw data unavailable. In this case, there is no way to estimate probability distribution. Nevertheless, we can invite some experts in the field to assess belief degree of each event. Belief degree means that you believe the possibility of the event. For example, we can obtain some imprecisely observed values by estimating the methane emissions. In order to deal with belief degree, Liu [7] established uncertainty theory in 2007. Based on uncertainty theory, the study of uncertain statistics began with Liu [8] in 2010 for estimating uncertainty distribution. Uncertain statistics contains two important tools in describing the imprecise observations: uncertain regression analysis and uncertain time series analysis.

Uncertain regression analysis is a statistical analysis method to determine the quantitative relationship between explanatory variables and response variables. The problem of parameter estimation is especially important in practice. There are many excellent methods such as least squares estimation ([19]), least absolute deviations estimation ([11]), Tukey’s biweight estimation ([1]), and maximum likelihood estimation ([6]). Moreover, Lio and Liu [5] gave a method for estimating the expected value and variance of uncertain disturbance term. Later on, Liu and Jia [9] applied cross validation to obtaining a reliable model.

Conventional time series analysis has attractive properties, but fails to deal with the imprecise observations. Uncertain time series analysis gets around that difficulty by describing imprecisely observed values with uncertain variables. Yang and Liu [16] have offered an elegant framework. An analytical result of the UAR model can be found in Zhao et al. [23]. Other methods of estimation include least absolute deviations estimation ([18]), Huber estimation ([12]), Lasso procedure ([22]), ridge estimation ([3]), and maximum likelihood estimation ([2]). In addition, Liu and Yang [10] used cross validation to determine the order of the UAR model. Practically, Ye and Yang [20] used the UAR model to analyze the cumulative confirmed cases of COVID-19 in China. Actual time series frequently contains outliers, for example, extremal events occur at certain time. In the framework of probability theory, the max-autoregressive model was explored by Davis and Resnick [4], and has been applied to the analysis of time series with outliers. Similarly, we fit the UMAR model to the imprecise observations with outliers. In many applications, other uncertain time series models are proposed, such as uncertain moving average model ([17]), uncertain vector autoregressive model ([14]), and uncertain threshold autoregressive model ([15]).

The remainder of the paper is structured as follows. Section 2 presents some foundational issues, such as the definition of the UMAR model. Section 3 provides the details of parameter estimation for the UMAR model. Residual analysis is shown in Section 4. Section 5 detects outliers via the residual analysis. Section 6 discuss the forecasting based on the UMAR model. In Section 7, the UMAR model is fit to China’s methane emissions data. Section 8 offers some concluding remarks.

2 Uncertain max-autoregressive model

In order to obtain a feel for the sample paths of the max-autoregressive model, Davis and Resnick [4] simulated a large number of observations. The result showed that the max-autoregressive graph is slightly smoother than the autoregressive graph. Accordingly, this section gives a brief description of the UMAR model. A formulation of the k-order UMAR model is given by X t = a 0 + ⋁ i = 1 k a i X t - i + ε t(1) where a₀, a₁, ⋯ , a _k are constants, and ε _t is a disturbance term.

The UMAR model exists simpler analytical structure than the UAR model [16], because the predicted value is only related to the maximum value in previous periods.

Given a set of imprecisely observed values X₁, X₂, ⋯, X _n , then we obtain the estimate of a₀, a₁, ⋯, a _k in model (1) by utilizing the least squares estimation in Yang and Liu [16], min a 0 , a 1 , ⋯ , a k ∑ t = k + 1 n E [ ( X t - a 0 - ⋁ i = 1 k a i X t - i ) 2 ] .(2)

Theorem 3. Let X₁, X₂, ⋯ , X _n be imprecisely observed values which can be regarded as independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. Then the least squares estimate of a₀, a₁, ⋯ , a _k in model (1) is found by solving the following mathematical programming min a 0 , a 1 , ⋯ , a k ∑ t = k + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - a 0 - ⋁ i = 1 k a i ϒ t - i - 1 ( α , a i ) ) 2 d α(3) where ϒ t - i - 1 ( α , a i ) = { Φ t - i - 1 ( 1 - α , a i ) , if a i ≥ 0 Φ t - i - 1 ( α , a i ) , if a i < 0 for i = 1, 2, ⋯ , k.

Proof. Theorem 1.26 in Liu [8] provides the operational law for calculating the inverse uncertainty distributions of uncertain variables. Then the inverse uncertainty distribution of X t - a 0 - ⋁ i = 1 k a i X t - i can be deduced as Ψ t - 1 ( α ) = Φ t - 1 ( α ) - a 0 - ⋁ i = 1 k a i ϒ t - i - 1 ( α , a i ) . Theorem 1.32 in Liu [8] provides the operational law for calculating the expected values of uncertain variables. Then we have E [ ( X t - a 0 - ⋁ i = 1 k a i X t - i ) 2 ] = ∫ 0 1 ( Ψ t - 1 ( α ) ) 2 d α . Top grid treatment thus for the uncertain programming problem (2) and mathematical programming problem (3), both solutions are identical. The proof is complete.

Showing the optimal solution of (2) with a 0 * , a 1 * , ⋯, a k * , then the fitted UMAR model is X t = a 0 * + a 1 * X t - 1 ∨ ⋯ ∨ a k * X t - k .

Top grid treatment in order to choose an appropriate order k, we utilize the fixed origin cross validation presented by Liu and Yang [10], ATE T ( k ) = 1 n - T ∑ t = T + 1 n E [ ( X t - a 0 * - ⋁ i = 1 k a i * X t - i ) 2 ] where a 0 * , a 1 * , ⋯ , a k * are the least squares estimate of a₀, a₁, ⋯ , a _k from training set {X₁, X₂, ⋯ , X _T } , and T is a given index with k < T < n. Then k is chosen to minimize the average testing error ATE _T  (k).

Top grid treatment according to Liu and Yang [10], ATE _T  (k) can be calculated by the following formula, ATE T ( k ) = 1 n - T ∑ t = T + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - a 0 - ⋁ i = 1 k a i ϒ t - i - 1 ( α , a i ) ) 2 d α .(4)

In this section, we focus on the estimation of the expected value and variance of disturbance term. Let X₁, X₂, ⋯, X _n be imprecisely observed values. According to Yang and Liu [16], ε ˜ t : = X t - a 0 * - ⋁ i = 1 k a i * X t - i is defined as the tth residual for each t (t = k + 1, k + 2, ⋯, n).

For model (1), ε_k+1, ε_k+2, ⋯, ε _n are assumed to be iid uncertain variables. Then the expected value of disturbance term can be estimated by the average of the expected values of residuals, i.e., e ˆ : = 1 n - k ∑ t = k + 1 n E [ ε ˜ t ] . Similarly, the variance of disturbance term can be estimated as σ ˆ 2 : = 1 n - k ∑ t = k + 1 n E [ ( ε ˜ t - e ˆ ) 2 ] .

Theorem 4. Let X₁, X₂, ⋯ , X _n be imprecisely observed values characterized in term of independent uncertain variables with regular uncertainty distributions Φ₁ (x) , Φ₂ (x) , ⋯ , Φ _n  (x), and let the fitted UMAR model be X t = a 0 * + ⋁ i = 1 k a i * X t - i . Then the estimated expected value of disturbance term is e ˆ = 1 n - k ∑ t = k + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - a 0 * - ⋁ i = 1 k a i * ϒ t - i - 1 ( α , a i * ) ) d α , and the estimated variance of disturbance term is σ ˆ 2 = 1 n - k ∑ t = k + 1 n ∫ 0 1 ( Φ t - 1 ( α ) - a 0 * - ⋁ i = 1 k a i * ϒ t - i - 1 ( α , a i * ) - e ˆ ) 2 d α where ϒ t - i - 1 ( α , a i * ) = { Φ t - i - 1 ( 1 - α ) , if a i * ≥ 0 Φ t - i - 1 ( α ) , if a i * < 0 for i = 1, 2, ⋯ , k.

Proof. Theorem 1.26 in Liu [8] shows that the inverse uncertainty distribution of ε ˜ t = X t - a 0 * - ⋁ i = 1 k a i * X t - i is Ψ ˆ t - 1 ( α ) = Φ t - 1 ( α ) - a 0 * - ⋁ i = 1 k a i * ϒ t - i - 1 ( α , a i * ) . By Theorem 1.32 in Liu [8], we have the following transformations, E [ ε ˜ t ] = ∫ 0 1 Ψ ˆ t - 1 ( α ) d α , and E [ ( ε ˜ t - e ˆ ) 2 ] = ∫ 0 1 ( Ψ ˆ t - 1 ( α ) - e ˆ ) 2 d α . The proof is complete.

5 Outlier detection

The term outlier refers to an observation which in some sense is inconsistent with the rest of the observations in the data set [13]. Outliers are produced in the process of observation or experiment due to unexpected events, which disturb the original statistical laws. They can be detected by the residuals. In Section 4, we obtain the residuals ε ˜ k + 1 , ε ˜ k + 2 , ⋯ , ε ˜ n with inverse uncertainty distributions Ψ ˆ k + 1 - 1 , Ψ ˆ k + 2 - 1 , ⋯, Ψ ˆ n - 1 , respectively. Assume disturbance term follows the normal uncertainty distribution N ( e ˆ , σ ˆ ) . Then the residuals can be regarded as n - k samples of 𝒩 ( e ˆ , σ ˆ ) . Large residuals, known as outliers, indicate possible problems with the original data.

According to Ye and Liu [21], we present the following outlier-detection procedure. Take a significance level α, (e.g., 0.05). If Ψ ˆ t - 1 ( 1 - α ) < Φ - 1 ( α ) or Ψ ˆ t - 1 ( α ) > Φ - 1 ( 1 - α ) where Φ^-1 (α) is the inverse uncertainty distribution of N ( e ˆ , σ ˆ ) , i.e., Φ - 1 ( α ) = e ˆ + σ ˆ 3 π ln α 1 - α , then ε ˜ t can be regarded as an outlier. Correspondingly, X _t is an outlier in X₁, X₂, ⋯ , X _n .

In some situations, the forecast is heavily influenced by outliers. Thus these outliers should be revised. If the observation X _i is an outlier, then one may try a transformation of data, such as X i = 1 2 ( X i - 1 + X i + 1 ) . The revised data X₁, X₂, ⋯, X _n can be used to re-estimate the fitted UMAR model and disturbance term until no more outliers are found. A better model is ultimately obtained.

6 Forecast value and confidence interval

This section presents a confidence interval to predict the value at the next time point. Based on Sections 3 and 4, we define a forecast uncertain variable X ˆ t : = a 0 * + ⋁ i = 1 k a i * X t - i + ε t . Assume ε_n+1 follows the normal uncertainty distribution N ( e ˆ , σ ˆ ) . By the independence of X _n , X_n-1, ⋯ , X_n+1-k and ε_n+1 (Theorem 1.34 in [8]), we obtain a forecast value of X_n+1, μ = E [ X ˆ n + 1 ] = E [ a 0 * + ⋁ i = 1 k a i * X n + 1 - i + ε n + 1 ] = a 0 * + ⋁ i = 1 k E [ a i * X n + 1 - i ] + E [ ε n + 1 ] = a 0 * + ⋁ i = 1 k a i * E [ X n + 1 - i ] + e ˆ . According to Theorem 1.26 in Liu [8], the inverse uncertainty distribution of X ˆ n + 1 is Φ ˆ n + 1 - 1 ( α ) = a 0 * + ⋁ i = 1 k a i * ϒ n + 1 - i - 1 ( α , a i * ) + Φ - 1 ( α ) where ϒ n + 1 - i - 1 ( α , a i * ) = { Φ n + 1 - i - 1 ( α ) , if a i * ≥ 0 Φ n + 1 - i - 1 ( 1 - α ) , if a i * < 0 for i = 1, 2, ⋯ , k. Taking α (e.g., 95%) as the confidence level, and finding the minimum value b such that Φ ˆ n + 1 ( μ + b ) - Φ ˆ n + 1 ( μ - b ) ≥ α . We suggest that the α confidence interval of X_n+1 is [ μ - b , μ + b ] .

7 Real data application

In this section, we use China’s methane emissions data (unit: thousand tons of carbondioxide equivalence) to demonstrate the application of the UMAR model. The original set of data, denoted by y₁, y₂, ⋯, y₄₂, can be shown in Table 1 and Fig. 1. In fact, methane emissions cannot be imprecisely observed. Hence, it is better to describe methane emissions with imprecisely observed values. Next, we suggest transformations on y₁, y₂, ⋯, y₄₂, and obtain the imprecisely observed values X₁, X₂, ⋯, X₄₂ described by independent linear uncertain variables with regular uncertainty distributions Φ₁ (x) , Φ₂ (x) , ⋯ , Φ₄₂ (x), respectively, as shown in Table 2 and Fig. 2, t = 1, 2, ⋯ , 42. Notice, X₃₅ (2004) is set to an outlier.

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Fig. 1

Raw precise observations.

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Fig. 2

Raw imprecise observations.

Considering the data of size n = 42, we choose T = 30 as the training set size. Based on Theorem 3 and the outlier-detection procedure in Section 5, the values of ATE₃₀ (2), ATE₃₀ (3) and ATE₃₀ (4) are calculated and listed in Table 3. The result is to choose k = 3, and the fitted UMAR model is X t = - 35355 + ( 1.06 X t - 1 ) ∨ ( 0.86 X t - 2 ) ∨ ( 0.96 X t - 3 ) . By Theorem 4, the expected value and standard deviation of disturbance term are estimated as e ˆ = 0.112 , σ ˆ = 45465 .

Denoting the inverse uncertainty distribution of N ( e ˆ , σ ˆ ) by Φ^-1, and taking significance level α = 0.05, we have Φ - 1 ( α ) = - 73808 , Φ - 1 ( 1 - α ) = 73808 . The inverse uncertainty distribution of the residual ε ˜ t = X t + 35355 - ( 1.06 X t - 1 ) ∨ ( 0.86 X t - 2 ) ∨ ( 0.96 X t - 3 ) is Ψ ˆ t - 1 ( α ) = Φ t - 1 ( α ) + 35355 - ( 1.06 Φ t - 1 - 1 ( 1 - α ) ) ∨ ( 0.86 Φ t - 2 - 1 ( 1 - α ) ) ∨ ( 0.96 Φ t - 3 - 1 ( 1 - α ) ) .

Fig. 3 displays a time series plot of Φ^-1 (0.95) and Φ^-1 (0.05), denoted by parallel red medium dashed lines, Ψ ˆ t - 1 ( 0.95 ) and Ψ ˆ t - 1 ( 0.05 ) , denoted by vertical black solid dashed lines. By the outlier-detection procedure in Section 5, ε ˜ 35 and ε ˜ 36 appear to be two severe outliers. Also, no further outliers are detected. It indicates that X₃₅ is an outlier in raw imprecisely observed values. Then the 35th imprecise value is revised to X 35 = 1 2 ( X 34 + X 36 ) . Based on the revised imprecisely observed values X₁, X₂, ⋯ , X₄₂, we obtain a new fitted model, X t = - 68935 + ( 1.09 X t - 1 ) ∨ ( 0.93 X t - 2 ) ∨ ( 0.84 X t - 3 ) . Top grid treatment the estimated expected value and standard deviation of disturbance term are e ˆ = - 0.208 , σ ˆ = 74150 , respectively, thereby Φ - 1 ( α ) = - 120381 , Φ - 1 ( 1 - α ) = 120381 . The inverse uncertainty distribution of the residual ε ˜ t = X t + 68935 - ( 1.09 X t - 1 ) ∨ ( 0.93 X t - 2 ) ∨ ( 0.84 X t - 3 ) is Ψ ˆ t - 1 ( α ) = Φ t - 1 ( α ) + 68935 - ( 1.09 Φ t - 1 - 1 ( 1 - α ) ) ∨ ( 0.93 Φ t - 2 - 1 ( 1 - α ) ) ∨ ( 0.84 Φ t - 3 - 1 ( 1 - α ) ) . Top grid treatment Fig. 4 displays the relationships between Ψ ˆ t - 1 and Φ^-1, t = 4, 5, ⋯, 42. It is obvious that there are not outliers, because the outlier in the raw data is revised well. Hence, the forecast uncertain variable is X ˆ t = - 68935 + ( 1.09 X t - 1 ) ∨ ( 0.93 X t - 2 ) ∨ ( 0.84 X t - 3 ) + ε t where ε t ∼ N ( 0 , 74150 ) . Using the forecast uncertain variable, we obtain the predicted value and 95% confidence interval of methane emissions in 2012 as follows, μ = 1776080 , ( 1626080 , 1926080 ) . The actual methane emissions in 2012 is 1752290 which falls into the 95% confidence interval.

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Fig. 3

The relationships between Ψ ˆ t - 1 and Φ^-1 with outliers, t = 4, 5, ⋯ , 42 .

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Fig. 4

The relationships between Ψ ˆ t - 1 and Φ^-1 without outliers, t = 4, 5, ⋯ , 42 .

Similarly, using the UAR model to fit the raw imprecisely observed values X₁, X₂, ⋯ , X₄₂, we can obtain a forecast uncertain variable X t = - 44660 + 1.27 X t - 1 - 0.13 X t - 2 - 0.08 X t - 3 + ε t where ε t ∼ N ( - 0.029 , 25510 ) . Fig. 5 displays the relationships between Ψ ˆ t - 1 and Φ^-1, t = 4, 5, ⋯, 42. Notice that there is no significant difference between Fig. 4 and Fig. 5. However, the latter has a smaller variance. Thus the UMAR model can achieve a satisfactory fit to the data, not just the UAR model.

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Fig. 5

The relationships between Ψ ˆ t - 1 and Φ^-1 from the UAR model, t = 4, 5, ⋯ , 42 .

This paper mainly gave a description of the UMAR model. Based on the principle of least squares, we obtained the estimates of unknown parameters in the UMAR model. The expected value and variance of disturbance term are estimated by the residuals. Furthermore, the residuals are used to discover outliers. The developed methods allowed us to consider the prediction problem as well. The application of the UMAR model was presented in an example. As for future research, more nonlinear uncertain time series models can be considered.