# Outlier Detection in Adaptive Functional-Coefficient Autoregressive Models Based on Extreme Value Theory

Authors: Ping Chen; Ling Dong; Wanyi Chen; Jin-Guan Lin
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper proposes several test statistics to detect additive or innovative outliers in adaptive functional-coefficient autoregressive (AFAR) models based on extreme value theory and likelihood ratio tests. All the test statistics  follow a tractable asymptotic Gumbel distribution. Also, we propose an asymptotic critical value on a fixed significance level and obtain an asymptoticp-value for testing, which is used to detect outliers in time series. Simulation studies indicate that the extreme value method for detecting outliers in AFAR models is effective both for AO and IO, for a lone outlier and multiple outliers, and for separate outliers and outlier patches. Furthermore, it is shown that our procedure can reduce possible effects of masking and swamping.


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## Body

## 1. Introduction

Outlier detection and analysis play important roles in practical applications. For instance, outlier detection can be applied to anomaly detection in computer networks, financial time series, and data series in geosciences, as can be seen from [1, Chap. 1] and [2]. Other examples include the study on loss of customers in the commercial field and the detection and tracking of financial crime as credit card fraud, all of which involve and exploit the useful information provided by the presence of outliers. On the other hand, outliers in dynamic systems or engineering time series [3] can have adverse effects on model identification and parameter estimation, where eliminating outliers is necessary in the statistical modeling of time series for the purpose of preprocessing data, see, for example, [4]. Some studies have even shown that the emerging of outliers generates certain nonlinear time series. Several procedures are available in the literature to deal with problems related to outliers. Chen [5] developed a method for detecting additive outliers in bilinear time series, which belong to the family of fractal time series models [6]. Cai et al. [7] studied the functional-coefficient regression models of nonlinear time series. Battaglia [8] discovered a way to identify and estimate outliers in functional autoregressive time series. Battaglia and Orfei [9] addressed the issue of outlier detection and estimation in nonlinear time series. Chen et al. [10] and Chen et al. [11] discussed the detection of outlier patches, change point, and outliers in bilinear models.Extreme value theory and likelihood ratio tests have been used in the detection of outliers and time series analysis. For instance, Martin [12] conducted extreme value analysis on the optimal level cross-prediction of linear Gaussian processes. Zhu and Ling [13] performed likelihood ratio tests on the structural change from an AR(p) model to a threshold AR(p) model. Furthermore, based on extreme value theory Chareka et al. [14] proposed an alternative test method for the detection of additive outliers in Gaussian time series. On the other hand, some scholars such as Fung et al. [15] and Río [16], focused their studies on special cases of outliers detection. It is commonly agreed that the key of outlier detection lies in determining whether the test statistic exceeds a critical value, that is, the threshold under a given significance level. However, explanations for the selection of threshold in many literatures are ambiguous, and the threshold itself can hardly be controlled under a certain level of significance. In this paper, we propose an asymptotic critical value on a fixed significance level, which is used to detect additive and innovative outliers in adaptive functional-coefficient autoregressive (AFAR) models (see, e.g., Fan and Yao [17]).This paper is structured as follows. In Section2, we consider AFAR models with additive or innovative outliers and their estimation. In Section 3, several procedures are proposed to detect outliers in the AFAR models based on extreme value theory and likelihood ratio tests. In Section 4, we present some simulation studies and demonstrate the effectiveness of the proposed method through an empirical way. Concluding remarks are summarized in Section 5.


## 2. Outliers Models and Test Statistics

We now consider the AFAR model:(1)xt=cos(a1xt-2)a2+a3xt-22xt-1+(a4+a5exp(a6xt-22))xt-2+εt,
which can be written as
(2)xt=∑j=12‍fj(xt-2)xt-j+εt,
where εt is a Gaussian white noise with mean zero and variance σ2 and f1(xt-2)=cos(a1xt-2)/(a2+a3xt-22), f2(xt-2)=a4+a5exp(a6xt-22).Suppose that{xt}t=1n is a zero-mean stationary process following model (2) description, and there is an outlier at time t=q, whose influence magnitude is ωq, then the observed series {yt} may be presented as follows.(1) Innovative outlier (IO) model:(3)yt=xt,1≤t<qyt=∑j=12‍fj(yt-2)yt-j+ηt,q≤t≤n,ηt=εt+ωqIO·δ(t-q).(2) Additive outlier (AO) model:(4)yt=xt+ωqAO·δ(t-q),t=1,…,n,
where δ(t) is the Kronecker symbol: If t=0, then δ(t)=1, else δ(t)=0.It is obvious that the detection for IO or AO may be transformed to test whether the null hypothesisH0:ωqIO=0 or H0′:ωqAO=0 is true under a certain level of significance. This can be solved by knowing the distributions of ω^qIO and ω^qAO under the null hypothesis. We estimate ωq by using the maximum likelihood method. The maximum likelihood function of observations is the positive ratio of the maximum likelihood function of residuals. For a given q, assume that the fj and corresponding parameters are known, then we can obtain the estimation of ωq by maximizing the likelihood function. For convenience, suppose that there is no outlier in the two former observations. Denote the residual of observations by ηt, for any k≤0, and let εk=0, xk=0, and fk(·)=0. ε1, ε2 be obtained by letting εt=xt-∑j=12‍fj(xt-2)xt-j. Under initial conditions mentioned above, the conditional likelihood function of ωq is given by
(5)l(y∣ω)=l(x)=l(ε)=∏t=3n(2πσ2)-1/2exp(-εt22σ2)=(2πσ2)-(n-2)/2exp(-∑t=3n‍εt22σ2).We can obtain the maximum likelihood estimation ofωq at the minimum of ∑t=3n‍εt2. It is an accurate estimation for the linear model. However, it is just an approximation for the nonlinear model. For t>2, the residual of observations is given by ηt=yt-∑j=12‍fj(yt-2)yt-j. We then discuss the estimations of ωqIO and ωqAO for model (2).(1) Assume that there is an IO att=q and its influence magnitude is ωqIO. From (3), it follows that ηq=εq+ωqIO and ηt=εt (t≠q), where ∑t=3nεt2=∑t≠qηt2+(ηq-ωqIO)2. So, the maximum likelihood estimation of ωqIO is given by
(6)ω^qIO=ηq=yq-∑j=12‍fj(yq-2)yq-j.
For known fj and σ2, we have ω^qIO~N(ωqIO,σ2), which is similar to Battaglia [8]. Under the null hypothesis H0:ωqIO=0, ω^qIO~N(0,σ2), we obtain likelihood radio test statistic λ^qIO=(ω^qIO/σ)~N(0,1) by standardizing ω^qIO. When fj and σ2 are unknown, they can be replaced by their consistent estimations σ^IO2=(n-2)-1∑t=3,t≠qn‍η^t2 and η^t=yt-∑j=12‍f^j(yt-2)yt-j. Similarly, we also have that
(7)λ^qIO=ω^qIOσ^IO~N(0,1).(2) Assume that there is an AO att=q and its influence magnitude is ωqAO. From (4), it follows that
(8)ηq=yq-∑j=12‍fj(yq-2)yq-j=xq+ωqAO-∑j=12‍fj(xq-2)xq-j=εq+ωqAO.
When t<q, ηt=yt-∑j=12‍fj(yt-2)yt-j=xt-∑j=12‍fj(xt-2)xt-j=εt, if t>q, that is, k>0, then ηq+k=yq+k-∑j=12‍fj(yq+k-2)yq+k-j, if k=1, we have that
(9)ηq+1=yq+1-∑j=12‍fj(yq-1)yq+1-j=xq+1-∑j=12‍fj(xq-1)(xq+1-j+ωqAOδ(1-j))=εq+1-∑j=12‍fj(xq-1)ωqAOδ(1-j)=εq+1-f1(xq-1)ωqAO=εq+1-f1(yq-1)ωqAO,
and if k=2, we have that
(10)ηq+2=yq+2-∑j=12‍fj(yq)yq+2-j=xq+2-∑j=12‍fj(xq)xq+2-j+∑j=12‍fj(xq)xq+2-j=-∑j=12‍fj(yq)yq+2-j=εq+2-(f1(yq)-f1(xq))yq+1=+f2(xq)(yq-ωqAO)-f2(yq)yq=εq+2-f1′(xq+θ1ωqAO)ωqAOyq+1=-(f2(yq)-f2(xq))yq-f2(xq)ωqAO=εq+2-f1′(xq+θ1ωqAO)ωqAOyq+1=-f2′(xq+θ2ωqAO)ωqAOyq-f2(xq)ωqAO=εq+2-ωqAO·[f2(xq)+∑j=12‍fj′(xq,j*)yq+2-j],
where xq,j*=xq+θjωqAO, 0<θj<1, and j=1,2. If k>2, then ηq+k=εq+k, from which it follows that
(11)ηq+k=ckωqAO+εq+k,k=1,2,
where c1=-f1(yq-1) and c2=-[f2(xq)+∑j=12‍fj′(xq,j*)yq+2-j], so we have that
(12)∑t=3n‍εt2=∑t=3q-1‍εt2+εq2+∑t=q+1q+2‍εt2+∑t=q+3n‍εt2=∑t=3q-1‍ηt2+(ηq-ωqAO)2+∑k=12‍(ηq+k-ckωqAO)2+∑t=q+3n‍ηt2=∑t=3q-1‍ηt2+∑t=q+3n‍ηt2+(1+c12+c22)ωqAO2=-2(ηq+c1ηq+1+c2ηq+2)ωqAO+(ηq2+ηq+12+ηq+22).
By minimizing the above expression, we obtain
(13)ωqAO=ηq+c1ηq+1+c2ηq+21+c12+c22,
where c1 may be obtained by estimation of -f1(yq-1), while c2 is complicated and difficult to be confirmed. Nevertheless, Battaglia [8] indicated that we could estimate c2 by -f2(yq), which is convenient and effective. If all fj(·) and σ2 are known, then we have
(14)ω^qAO=ηq-f1(yq-1)ηq+1-f2(yq)ηq+2Δq,Δq=1+f12(yq-1)+f22(yq).
Similar to Battaglia [8], we obtain
(15)ω^qAO~N(ωqAO,σ2Δq).
Under the null hypothesis H0′:ωqAO=0, we obtain likelihood radio test statistics λ^qAO=(ω^qAO/(σ/Δq))~N(0,1) by standardizing ω^qAO. In practice, if fj(·) and σ2 are unknown, which can be replaced by their consistent estimations σ^AO2=(n-2)-1[∑t=3n‍η^t2-(ω^qAO)2·∑j=12‍f^j2(yq+j-2)], we likewise obtain
(16)λ^qAO=ω^qAOσ^AO/Δ^q~N(0,1).It is indicated by (6) and (13) that the best estimate of ω is ηq at time q for the case of IO. the best estimate of ω is the linear combination of errors ηq, ηq+1, and ηq+2 at time q in the case of AO. If the location q of outlier is known, the distributions of likelihood ratio test statistics λ^qIO and λ^qAO are given by (7) and (16). However, in practice the location of outlier is unknown, thus we should inspect the magnitude of test statistic at every time point. By replacing λ^qIO and λ^qAO with λ^tIO and λ^tAO, the following theorem indicates that the processes of {λ^tIO} and {λ^tAO} are both Gaussian process, respectively.Theorem 1.
(1) Under the hypothesisH0:ωtIO=0, that is, there is no IO at any time, {ω^tIO} is a stationary zero-mean Gaussian process with variance σ2, and its autocovariance function is rIO(h)=σ2·I(h=0), where IA(x) is the index function. If x∈A, then IA(x)=1, and else IA(x)=0.
(2) Under the hypothesisH0′:ωtAO=0, that is, there is no AO at any time, {ω^tAO} is a stationary zero-mean Gaussian process with variance σ2/Δt, and its autocovariance function is
(17)rAO(t,h)=σ2Δt-1Δt+h-1[∑j=h+12‍fj(yt+j-2)fj-h(yt+j-2)σ2Δt-1Δt+h-1m-fh(yt+h-2)∑j=h+12‍]I(1≤h≤2).Proof.
The Gaussian character is shown by (7) and (16). We derive their autocovariance function as follows.(1)
Under the null hypothesisH0:ωtIO=0, we have ηt=εt and
(18)rIO(h)=cov(ω^tIO,ω^t+hIO)=cov(ηt,ηt+h)=cov(εt,εt+h)=σ2·I(h=0).(2)
Under the null hypothesisH0′:ωtAO=0, we have ηt=εt and
(19)rAO(t,h)=cov(ω^tAO,ω^t+hAO)=cov(Δt-1(ηt-∑j=12‍fj(yt+j-2)ηt+j),=covkkΔt+h-1(ηt+h-∑j=12‍fj(yt+h+j-2)ηt+h+j))=Δt-1Δt+h-1[∑j=12‍cov(ηt,ηt+h)=Δt-1Δt+h-1m-∑j=12‍fj(yt+h+j-2)cov(ηt,ηt+h+j)=Δt-1Δt+h-1m-∑j=12‍fj(yt+j-2)cov(ηt+j,ηt+h)=Δt-1Δt+h-1m+cov(∑j=12‍fj(yt+j-2)ηt+j,=Δt-1Δt+h-1m+covkk∑j=12‍fj(yt+h+j-2)ηt+h+j)]=σ2Δt-1Δt+h-1[∑j=h+12‍fj(yt+j-2)fj-h(yt+j-2)=σ2Δt-1Δt+h-1m-fh(yt+h-2)∑j=h+12‍]I(1≤h≤2).We know that{ω^tIO} is not only Gaussian, but also a stationary process by Theorem 1. Also, it is serial independent under the null hypothesis H0:ωtIO=0. However, despite that {ω^tAO} is Gaussian, it is not stationary due to the fact that its autocovariance function rAO(t,h) is dependent on t. Besides, {ω^tAO} is truncated in finite steps, which means that its maximum correlation is within two steps; that is, ω^tAO is independent with ω^t+hAO when h>2. By standardizing {ω^tIO} and {ω^tAO}, we can obtain {λ^tIO} and {λ^tAO}. It is obvious that {λ^tIO} is a standardized stationary Gaussian process under the hypothesis H0:ωtIO=0. Also, {λ^tAO} is a standardized Gaussian process under the hypothesis H0′:ωtAO=0, but it is not stationary, whereas its autocovariance function is truncated in finite steps.


## 3. Detection of Outliers Based on Extreme Value Theory

As similar to Chareka et al. [14] and Leadbetter and Rootzén [18], we obtain the following.Lemma 2 (see [18]).
Assume that{Xn} is a stationary zero-mean Gaussian time series and its autocorrelation function is ρ(h). Let Zn=max{X1,X2,…,Xn}, and if the Berman condition limh→∞ρ(h)·logh=0 is satisfied, then one has that P(((Zn-cn)/dn)≤x)→Λ(x)=exp{-e-x}, n→∞, where dn=(2logn)-1/2, and cn=(1/dn)-(dn/2)(loglogn+log(4π)).Lemma 3 (see [18]).
Assume that{Xn} is a stationary zero-mean Gaussian time series, and let An=maxt=1n{|Xt|} and Z2n=maxt=12n{Xt}, and if Xn satisfies the Berman condition, then An and Z2n have the same distribution.Theorem 4.
LetηIO=maxt=1,2,…,n{|λ^tIO|} and ηAO=maxt=1,2,…,n{|λ^tAO|}, and under the hypothesis H0:ωtIO=0 and H0′:ωtAO=0, one has that
(20)P(ηIO-c2nd2n≤x)⟶Λ(x),P(ηAO-c2nd2n≤x)⟶Λ(x),n⟶∞,
where d2n=[2log(2n)]-1/2 and c2n=(1/d2n)-(d2n/2)[loglog(2n)+log(4π)].Proof.
It follows from Theorem1 that the autocorrelation function of {λ^tIO} is ρIO(h)=I(h=0) under the hypothesis H0:ωtIO=0, so limh→∞ρIO(h)·logh=0; that is, the Berman condition is satisfied. Again, {λ^tIO} is a stationary zero-mean Gaussian time series. Hence, from Lemma 2, we have that P(((maxt=1n{λ^tIO}-cn)/dn)≤x)→Λ(x). Again from Lemma 3, we know that the distribution of ηIO=maxt=1n{|λ^tIO|} is the same as that of maxt=12n{λ^tIO}, so we have that P(((ηIO-c2n)/d2n)≤x)→Λ(x). Similarly, under the hypothesis H0′:ωtAO=0, from Theorem 1 we know that the autocorrelation function of λ^tAO is
(21)ρAO(t,h)=Δt-1/2·Δt+h-1/2·[∑j=h+12‍fj(yt+j-2)fj-h(yt+j-2)-fh(yt+h-2)∑j=h+12‍]I(1≤h≤2),
where limh→∞ρAO(h)·logh=0; that is, the Berman condition is satisfied. Therefore, similar to the statements above, it must hold that P(((ηAO-c2n)/d2n)≤x)→Λ(x).Lemma 5 (see [14]).
Assume that{Xn} is a stationary zero-mean Gaussian time series with variance 1 and autocorrelation function ρ(h), and let Mn=max{X12,…,Xn2}, and if ρ(h) satisfies the Berman condition, then one has P(((Mn-en)/2)≤x)→Λ(x), n→∞, where en=2logn-log(logn)-logπ.Theorem 6.
LetϑIO=maxt=1n{(λ^tIO)2} and ϑAO=maxt=1n{(λ^tAO)2}, under the hypothesis H0:ωtIO=0 and H0′:ωtAO=0, and if the corresponding Berman conditions for {λ^tIO} and {λ^tAO} are satisfied, then one has
(22)P(ϑIO-en2≤x)⟶Λ(x),P(ϑAO-en2≤x)⟶Λ(x),n⟶∞,
where en=2logn-log(logn)-logπ.Proof.
We know that the Berman conditions for{λtIO} and {λtAO} are satisfied under the hypothesis H0:ωtIO=0 and H0′:ωtAO=0 by the inferential process of Theorem 4. Thus, the conclusion is also obtained by Lemma 5.Letα be the test significance level and yn=(1+(1-α)1/n)/2. We denote the t-distribution function with n degrees of freedom by Hn(t). Let xn=Hn-2-1(yn) and e¯n=(n/(n-1))xn2-2Λ-1(1-α), where Λ-1(1-α) is the 1-α quantile of the Gumbel distribution. Similarly, we have the following.Theorem 7.
Under the conditions above, one has that(23)P(Mn-e¯n2≤x)⟶Λ(x),P(ϑIO-e¯n2≤x)⟶Λ(x),P(ϑAO-e¯n2≤x)⟶Λ(x),n⟶∞.At this point, it is convenient to introduce two more pieces of notation:AtIO=((|λ^tIO|-c2n)/d2n) and AtAO=((|λ^tAO|-c2n)/d2n), which we call absolute value test statistics; StIO=(((λ^tIO)2-en)/2) and StAO=(((λ^tAO)2-en)/2), which we call square test statistics; CtIO=(((λ^tIO)2-e¯n)/2) and CtAO=(((λ^tAO)2-e¯n)/2), which we call adjusted square test statistic. We likewise obtain
(24)P(maxt=1n{AtIO}≤x)⟶Λ(x),P(maxt=1n{AtAO}≤x)⟶Λ(x),P(maxt=1n{StIO}≤x)⟶Λ(x),n⟶∞,P(maxt=1n{StAO}≤x)⟶Λ(x),P(maxt=1n{CtIO}≤x)⟶Λ(x),P(maxt=1n{CtAO}≤x)⟶Λ(x),n⟶∞.If there are several outliers, the main idea is to first detect the maximum outlier by using our method and to obtain new test statistics by deleting its effect. Then, we go on to detect the next outlier and repeat the procedure till there is no outlier. When two types of outliers appear, the test is(25)H0:ωtIO=0,ωtAO=0versusH1:ωtIO≠0orωtAO≠0,0,ωtAO=0versusH1:ωtIO≠0orωtAO≠0t=1,…,n.For details, we provide the following steps using absolute value test statistics to detect the AO and IO in AFAR model as follows: (a) Let test statisticsT1=maxt=1n{AtIO} and T2=maxt=1n{AtAO}. (b) Calculate λ^tIO and λ^tAO at every time point, and calculate the maximum t1=Aq1IO=maxt=1n{AtIO} and t2=Aq2AO=maxt=1n{AtAO} of absolute value test statistics. (c) Let CV=Λ-1(1-α)=-log(-log(1-α)), if t1=max{t1,t2}>CV, then we believe that the observation is an IO at t=q1, and if t2=max{t1,t2}>CV, then we believe that the observation is an AO at t=q2; else, we believe there is no IO or AO. (d) Calculate the p-value p1=PH0(T1≥t1)=1-Λ(t1) and p2=PH0(T2≥t2)=1-Λ(t2), and if p1≤α or p2≤α, then reject the hypothesis H0, believing there is an IO or an AO. Furthermore, we decide whether it is an IO or an AO by its minimal p-value. (e) Delete the effect of detected outliers and detect the next outlier. Repeat the above steps till there is no outlier.


## 4. Simulation Studies

Example 8.
In this simulation, we consider the following model, where there is only one IO. The sample sizes aren=50, 100, 500, 1000, respectively and ωn/2IO=4. We have
(26)yt=cos(1.5yt-2)1+yt-22yt-1+(-exp(-yt-222))yt-2+ηt,ηt=εt+ωn/2IO·δ(t-n2),εt~N(0,0.5).
Forn=50, All the three test statistics detect an outlier at t=25. The p-values for believing it is an IO are p(AIO)=4.8414e-6, p(SIO)=1.8326e-8, and p(CIO)=3.3739e-8, respectively. The p-values for believing it is an AO are p(AAO)=3.1509e-4, p(SAO)=4.7392e-5, and p(CAO)=8.7247e-5, respectively. Because the p-values for believing it is an IO are smaller than the p-values for believing it is an AO, we believe it is an IO, and its influence magnitude is 3.1994. The results are similar for n=100, 500, 1000, respectively, we thus omit the details here. For different test statistics, their p-values for believing it is an IO are summarized as follows in Table 1.Table 1
p-values
Absolute value statistic
Square statistic
Adjusted square statistic
n
=
50
4.8414
e
-
6
1.8326
e
-
8
3.3739
e
-
8
n
=
100
1.0027
e
-
6
1.6592
e
-
9
2.1995
e
-
9
n
=
500
1.9065
e
-
6
2.6461
e
-
8
2.60394
e
-
8
n
=
1000
2.8048
e
-
7
1.3358
e
-
9
1.2590
e
-
9Example 9.
There are four IOs and one AO in the time series. These IOs appear att=49, t=50, t=51, and t=52 sequentially. Their sizes are 3, 6, 5, and 4, respectively. One AO appears at t=70 alone, and its size is 7, n=100. The model is as follows:
(27)xt=cos(1.5xt-2)1+xt-22xt-1+(-exp(-0.5xt-22))xt-2+εt,εt~N(0,0.6),
where yt is obtained by (3) or (4).
First, we detect an AO att=70, and its size is 6.8904. The three test statistics and their corresponding critical values are shown in Figure 1, where Figure 1(a) corresponds to the absolute value test statistics, Figure 1(b) corresponds to square test statistics, and Figure 1(c) corresponds to adjusted square test statistics. Symbol ‘‘⋄’’ denotes IO, and ‘‘*’’ denotes AO. Parallel broken line denotes critical value, which is uniform in the figures. Deleting the effect of this AO and then continuing to detect other outliers in the series, we observe an IO at t=50, whose size is 6.0285; see details in Figure 2. Deleting the effect of the aforementioned two outliers and then continuing to detect other outliers in the series, we observe an IO at t=51, whose size is 4.8119. Deleting the effect of the aforementioned three outliers and then continuing to detect other outliers in the series, we observe an IO at t=52, whose size is 4.0024. Deleting the effect of the aforementioned four outliers and then continuing to detect other outlier in the series, we detect an IO at t=49, whose size is 2.9046; see Figure 3. Deleting the effect of the aforementioned five outliers and then continuing to detect other outliers in the series, we have not detected any outlier; see Figure 4. Considering the length of paper, we omit some figures here. The result is consistent with the advance enactment.Example9, first detection.
(a)
(b)
(c)Example9, second detection.
(a)
(b)
(c)Example9, fifth detection.
(a)
(b)
(c)Example9, sixth detection.
(a)
(b)
(c)


## 5. Conclusions

The FAR model is mainly featured by the model-dependent-variable, which in one way or another limits the scope of its applications. As a generalization of the class of models, AFAR model clearly covers a larger range of objects than the FAR model, which makes it possible to reduce modeling biases [17] via choosing a proper model-dependent direction. This paper is concerned with detecting AO and IO in AFAR models using extreme value methods. We derive the asymptotic distribution of test statistics and provide a control for significance level, which serves as an extension and improvement of existing methods. Based on several simulation studies, we give conclusion remarks as follows. (a) The extreme value method for detecting outliers in AFAR models is tractable and effective not only for IO and AO, but also for separate outliers and outlier patches. Furthermore, it is shown that our method can reduce possible effects of masking and swamping. (b) When applying extreme value theory to detect outliers with a relatively small size samples at hand, the employment of square test statistics works better than that of adjusted square test statistics as well as absolute value test statistics. While following the increments in the samples size, the detecting effect of adjusted square test statistics also increases against that of square test statistics and absolute value test statistics (e.g., see Table 1). (c) Selection of model parameters and the magnitude of outliers have huge influences on the effect of detection.



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