**Professor Ke-Sheng Cheng (Email: rslab@ntu.edu.tw)**

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University

The following problems are intended for student groups KSC-1 and KSC-2.

KSC-1: 鄭梓辰 林吉堃 李安勳 林楷傑 林宏彥 張文豪 (Problem-1)

KSC-2: 蔡旻臻 李宜臻 陳裕翔 劉海柔 陳嘉聰 凌家宜 (Problem-2)

**Peoblem-1. Do we conduct hydrological frequency analysis correctly?**In many studies of hydrological analysis and environmental risk assessment, extreme values (i.e., quantiles of low exceedance probabilities) are of major concern. For example, annual maximum series (or block maximum series) are widely used in hydrological frequency analysis. Observed data in the annual maximum series are assumed

**IID**() and thus can be used to estimate the corresponding quantiles of different exceedance probabilities (or return periods). In Taiwan, almost all annual maximum rainfalls of longer durations (for example, 24-hr) are drawn by typhoons and annual counts of typhoons vary among years.*independently and identically distributed*In addition, in analogy to the Central Limit Theorem (CLT) which states that the sample mean is asymptotically normally distributed, the block maximum has an asymptotic distribution of the generalized extreme value distribution (GEV) family. This is known as the Extremal Types Theorem (ETT).

From the above description, the following questions and comments have been raised:

Is the IID a valid assumption for the annual maximum series?

Although it has been proved that block maxima are asymptotically GEV distributed, in many applications (most studies in Taiwan) of hydrological frequency analysis, GEV was not chosen to characterize the probability distribution of the annual maximum series. These applications seemed to contradict the Extremal Types Theorem and thus their quantile estimates might not be accurate.

**Week-1 (9/29) Topics to be discussed in class:**(1) 隨機樣本與統計參數推估

(2) 年最大值數列(Annual Maximum Series)與順序統計量(Order Statistics)

(3) Central Limit Theorem (CLT) and Extremal Types Theorem (ETT)

(4) Poisson process and gamma distribution

請各位同學先就上述各項內容，閱讀或複習相關定義與理論，以利課堂討論。

References: Data Analysis in Extreme Value Theory

**Problem-2. A simple classification problem with consideration of uncertainties**Classification of objects on the basis of certain features is fundamental in many scientific studies including taxonomy, remote sensing, hydrology, climatology, computer pattern recognition, etc. An error matrix (or confusion matrix) is used to evaluate the classification results. However, various forms of uncertainty in classification results are often ignored. In this problem, we will consider a simplest case of classification, i.e., one classification feature and two classes (categories), and look into details of uncertainty assessment and interpretation of the confusion matrix.

**Week-1 (9/29) Topics to be discussed in class:**(1) Maximum likelihood classifier

(2) Confusion matrix (error matrix)

(i) Training-data-based

(ii) Reference-data-based

(3) A simplest case of classification (one classification feature, two categories)

請各位同學先就上述各項內容，閱讀或複習相關定義與理論，以利課堂討論。

References: ML and Bayes Classifiers

**RSLAB - NTU**

**Prof. Ke-Sheng Cheng **

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University