This practical data are the wind speed data from October 2017 to September 2018, recording the maximum wind speed per hour. We apply two methods to this data to see if we can classify the different maximum wind speed trends in different seasons.
Here we treat one month of data as a curve with a total of 12 curves which are shown as figure 5.1. The first curve is the wind speed data for October 2017, the second curve
is the wind speed data for November 2017, and so on. The last curve is the wind speed data for September 2018. Table 5.1 shows the mean and standard deviation of monthly wind speed data. It shows that from October 2017 to February 2018 have larger wind speed, and from May 2018 to August 2018 have smaller standard deviation.
Table 5.1: The mean and standard deviation of monthly wind speed data
(mm/yy) 10/17 11/17 12/17 1/18 2/18 3/18 4/18 5/18 6/18 7/18 8/18 9/18 mean 14.95 15.04 16.87 14.64 14.26 9.67 7.42 6.78 8.53 6.49 6.97 9.89
sd 7.27 6.12 6.85 5.20 6.33 5.86 5.80 3.84 4.84 3.48 3.50 7.32
We firstly use the mixture of regression model to group them. According to the BIC, the number of selected clusters is three, and the estimated parameters β of each group are as Table 5.2. Based on the estimated values in Table 5.2, weˆ
draw the regression line in Figure 5.2. It can be seen from this figure that the different properties of the three groups are low wind speed, increasing wind speed and decreasing wind speed. Cluster 1 is the decreasing wind speed that contains the wind speed of 11/2017, 12/2017, and 2/2018. Cluster 2 is the low wind speed that contains the wind speed of March to August in 2018. Cluster 3 is the increasing wind speed that contains the wind speed of 10/2017, 1/2018, and 9/2018. Secondly, we use the hierarchical
Table 5.2: The estimated parameters ˆβ of each group Cluster 1 Cluster 2 Cluster 3 coef.(Intercept) 17.648602 7.173842 8.766866 coef.x -0.008070 0.015417 -0.002302 coef.x2 0.000015 0.000008 -0.000006
Figure 5.2: Data and regression lines by estimated values
clustering with DTW to group them. First we get the dendrogram as shown in Figure 5.3. From this dendrogram, the data is divided into three groups. The group on the left is from December 2017 to February 2018. The middle group is March to August 2018. The group on the right is October and November of 2017 and September of 2018.
Table 5.3: The clustering results obtained by the two methods
(mm/yy) 10/17 11/17 12/17 1/18 2/18 3/18 4/18 5/18 6/18 7/18 8/18 9/18
MRM 3 1 1 3 1 2 2 2 2 2 2 3
DTW 1 1 2 2 2 3 3 3 3 3 3 1
In the Table 5.3, we compare the clustering results obtained by the two meth-ods. According to Table 5.3, March to August are divided into groups by both two methods, but the result in other months are slightly different. When grouping by mixture of regression models, November 2017, December 2017 and February 2018 are divided into the same group, and January 2018, September 2018 and October 2017 are divided into the same group. However, when grouping by the hierarchical clustering, November 2017 is divided into the same group as October 2017 and September 2018, and January 2018 is divided into the same group as December 2017 and February 2018.
According to this result, we find out that the result of hierarchical clustering with DTW is similar to the average, and it also shows that the maximum wind speed have different behaviors in different seasons.
Chapter 6 Conclusions
In this thesis we compare two clustering methods for curves data. First, we consider different numbers of model component and use EM-algorithm to estimate pa-rameters. Second, we consider hierarchical grouping where DTW is used to calculate the distance between two curves. We compare these two methods by the cluster cor-recting rate.
By the result of simulation study, we find out that when the variation of curves is small, hierarchical clustering with DTW get a better result. However, when the vari-ation of curves is large, mixture of regression models is a better choice. We also use these two methods to analyze a practical data. According to the results, we find out that the clustering results of hierarchical clustering with DTW are more in line with our expectations. The behavior of maximum wind speed will be different in different seasons.
There are still many problems that we can discuss in the future. First, in
mix-to successfully divide the data inmix-to the number of groups we specify. Second, we choose algorithm to estimate parameter, there are also classification version of the EM-algorithm (CEM) (Celeux & Govaert, 1992) and stochastic version of the EM-EM-algorithm (SEM) (Celeux, 1985) can apply and discuss. We can also consider partitional clus-tering, for example, K mean or fuzzy C mean, with DTW distance. Third, the curves we consider in this thesis is one-dimensional. We can also discuss two-dimensional or above. Fourth, we consider the points in each curve are independent, the dependent situation also could be discussed in the future. Finally, the maximum wind speed in the practical analysis can be considered using the AR(1) model to analyze.
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