This document summarizes a seminar presentation on detecting outliers using the Ueda's method. The presentation defined outliers, described Ueda's methodology which uses the Akaike Information Criterion to identify outliers. It provided steps to calculate the test statistic Ut and interpret the results. Examples were shown applying the method to simulated normally distributed data with outliers added. The conclusion noted the method focuses on distribution symmetry and is effective for large samples.

1. Yashwantrao Chavan Institute of Science,
Satara
Department of Statistics
M.Sc.1 2017-2018
Seminar on
Automatic detection of discordant
outliers via the Ueda’s method
Marmolejo-Ramos et al. Journal of Statistical Distributions and
Applications (2015) 2:8
Presented by,
Patil Pooja Rajaram
Roll No. 115

2. Content:
 Introduction
 What is an outlier?
 Methodology
 Procedure
 How to interpret the results
 Examples
 Conclusion
 References

3. Introduction:
The importance of identifying outliers in a data set is
well known. Although various outlier detection methods have
been proposed in order to enable reliable inferences regarding
a data set, a simple but less known method has been proposed
by Ueda. the method proposed by Ueda which still assumes
the underlying data to be normally distributed and based on
the Akaike’s Information Criterion (AIC), presents interesting
features.
The aim of this paper is to study the performance and
robustness of the Ueda’s method to detect outliers via
computer simulations, as well as determine its applicability to
other types of data.

4.  What is an Outlier?
An outlier is an observation that lies an
abnormal distance from other values in a random
sample from a population. Outliers are generally
viewed as data values which are numerically
distant from the bulk of the sample.

5. Importance of identifying outliers:
By definition, outliers are points that are
distant from remaining observations. As a result
they can potentially skew or bias any analysis
performed on the data set. Thus it is very
important to detect and adequately deal with
outliers.

6.  Methodology:
Let X be a random variable with probability distribution f(x), and let
x1, x2, . . . , xn, xn+1, . . . , xn+s−1, xN be a random sample of size N = n + s from this
distribution, with n and s = {1, 2, 3, . . .} being the number of regular and outlying
observations, respectively.
The Ueda’s method is a simplified version of the use of the Akaike
Information Criterion (AIC).
AIC = 2k − 2 ln(L), (1)
In the expression above, k is the number of independently adjusted parameters and
L is the maximum likelihood value for n number of regular observations. In the
normal distribution, the AIC becomes:
AIC = 2n ln σ − 2 lnn!+2s (2)
Ueda used the correction factor; 2𝑠
𝑙𝑛𝑛!
𝑛
which still depends on n, but includes the
number of outliers s in the full sample. Based on this, the proposed test statistic to
identify outliers using this method is given by,
Ut =
1
2
AIC
≈ nlnσ - 2𝑠
𝑙𝑛𝑛!
𝑛
=(N-s) lnσ - 2𝑠
𝑙𝑛𝑛!
𝑛
(3)

7.  CalculatingUt :
the following procedure is suggested to determine outliers in a
full sample:
1. Order the full sample to obtain xord =x(1), x(2), . . . , x(N), with x(i) being
the i th order statistic (i = 1, 2, . . . ,N).
2. Calculate the z -score for each x(i) as,
Z(i) =
𝑥 𝑖− 𝑥
𝑠
Where, 𝑠 = 𝑖=1
𝑁
(𝑥𝑖− 𝑥)
𝑁−1
3. Provided a number of steps s ≥ 0, calculate the test statistic Ut as in
Eq. (3).Observe that s = 0 means that no outliers are present in the data.
Furthermore, as the original sample is ordered, s > 0 also implies that
the outliers are at either end of the sample.
4. Suppose that s = 1. It could be the case that the outlier is located
either at the beginning of the ordered sample, i.e. x(1) is the outlier or at
the end of it, i.e. x(N) is the outlier. If the former, then the test statistic Ut
is calculated as in Eq. (3) after removing x(1) from the ordered sample.
If the latter, x(N) is removed and Ut subsequently calculated.
5. Step 4 continues for other values of s > 1. As a result of this process,
a collection of Ut values is obtained.

8.  Interpretingthe Ut statistic:
The previous section described how the test statistic Ut is
calculated provided a value of s≥0. One question that remains to be
answered is how many values of Ut need to be calculated. Let C be such a
number and suppose that up to s outliers are to be detected in the sample.
Then, from the collection of Ut values obtained in step 5, the following
matrix is constructed:
Ut =
𝑈00
𝑈10
𝑈20
⋮
𝑈 𝑚0
𝑈01
𝑈11
𝑈21
⋮
𝑈 𝑚1
𝑈02
𝑈12
𝑈22
⋮
𝑈 𝑚2
⋯
⋯
⋯
⋯
⋯
𝑈0𝑚
𝑈1𝑚
𝑈2𝑚
⋮
𝑈 𝑚𝑚
Where m=s+1, U00 is Ut statistic when no outliers are detected
and Uij is the Ut statistic when i and j outliers are detected in the lower
and upper tails of xord. From Ut it is clear that the number of calculations
to detect up to s outliers in xord is atmost 𝑚2.
However, this number can be reduced to 𝑚2 − 2 if U00 and
Umm, included for convenience and comparison purposes, are not
calculated. Hence, 𝑚2
− 2 < 𝑐 < 𝑚2
.

9.  Example 1:
Data (sample size = N = 10): 2.02, 2.22, 3.04, 3.23, 3.59,
3.73, 3.94, 4.05, 4.11, 4.13
 Conclusion:
Here, first two observations are outliers hence
they should be removed from data.

10. • Example 2:
Situation in which no outlier is present and Ut obtained:
Data (sample size = 10): 5.4, 5.4, 5.5, 5.7, 5.8, 5.9, 6.0, 6.1,
6.3, 6.4
• Conclusion:
From the above table, we conclude that there do
not present any outlier in given data.

11. Example 3 : (Using simulated data)
Normally distributed data.
Consider a random sample x1, x2, . . . , x25 from a
N(300, 102) distribution, and a second random sample
y1, y2, . . . , y5 of outlier observations from a N(400, 52)
distribution. Then, the full sample of size N = n + s = 30
will be given by x1, x2, . . . , x25, y1, y2, . . . , y5.
Although the true number of outliers is s = 5,
smax is set to 10 so up to 10 observations are automatically
tested for their potentiality to be outliers using the Ueda’s
method. As shown in Fig.a, the number of outliers
detected in the full sample is s = 5.

12. sObservations
• Conclusion:
the outlying observations were sampled from normal
distributions and placed on the right tail of the normal
distribution.
• Graphical representation of example 3:

13. • Conclusion:
The Ueda’s method is an outlier detection method
that focuses on the symmetry and skewness of the
distribution in order to detect outlying data points. This
method is highly sensitive to outliers when the distribution
under analysis is negatively-, positively-skewed or
asymmetric about the centre of the distribution and such
sensitivity is enhanced by an increase in sample size.
• The advantages of the Ueda’s method are that:
i) It is easy to calculate.
ii) It does not require determining the number of potential
outliers in advance.
iii) It does not depend on tables or charts.
iv) It effectively signals the absence of outliers, and
v) It can be used with large sample sizes.

14. References :
Automatic detection of discordant outliers via
the Ueda’s method
(Marmolejo-Ramos et al. Journal of Statistical
Distributions and Applications (2015) 2:8)

15. THANk
YOU ……