Yeni zelanda GPS zaman serileri verisinin bayesci istatistik ile incelenmesi
Investigation of the New Zealand time series data with bayesian statistics
Yazar: KUBİLAY ÖZCAN
Danışman: PROF. DR. GÜRSEL SUNAL ; PROF. DR. MEHMET SİNAN ÖZEREN
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Jeoloji Mühendisliği Ana Bilim Dalı / Jeoloji Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2023
Konu: Jeoloji Mühendisliği = Geological Engineering

Özet:
İstatistik dünyasında yıllar boyunca birçok veri işleme, stokastik süreçleri anlama
yöntemleri gelişmiştir. Büyük veriler üzerinde kullanılması zor olan bu yöntemlere
teknolojinin gelişimi sayesinde getirilen yaklaşımsal ek yöntemler, bu yöntemlerin
büyük veriler üzerinde de kullanılmasına olanak sağlamıştır. Bu çalışmada Yeni
Zelanda'dan alınan GPS (Küresel Konumlama Sistemi) zaman serisi veri seti
bilgisayar tabanlı Bayesci istatistik yöntemleri ile incelenmiştir.
Çalışma için Yeni Zelanda'nın Jeolojik ve Nükleer Bilimler Enstitüsü'nden (GNS
Sciences) alınan, 2011-2021 yılları arasındaki, günlük mikrometre (mikron; 10-6 m)
mertebesinde ölçüm yapılan 146 GPS istasyonlu GPS yer hareketleri veri seti
kullanılmıştır. Bu veri seti üzerinde eksik veri noktaları Beklenti Maksimizasyonu
algoritması ile doldurulmuştur. Zaman serilerinde mevcut olan trend bileşeni yüksek
korelasyona neden olarak analizi zorlaştırdığı için zaman serileri de-trend edilmiştir.
Çalışmada doğrusal regresyon yapılmak istenmiştir. Ancak veri setinin büyük olması
nedeniyle olası 280 tane regresyon modeli olduğundan problem klasik regresyon
analizi ile çözülememektedir. En makul regresyon modelinin teşhisi için Stokastik
Arama ile Değişken Seçimi yapılmış bunun için Markov Zincirleri tabanlı bilgisayar
Gibbs örneklemesi algoritması kullanılmıştır. Markov zinciri 30000 zincirden
oluşacak şekilde tasarlanmıştır. Markov zincirinde potansiyel makul tahmin ettirici
değişkenlerin regresyon modelindeki katsayıların, Bayesci istatistiğin de bir kavramı
olan, önsel dağılım için "Spike-and-Slab Prior" olarak bilinen bir dağılım modeli
önerilmiştir. Oluşturulan Markov zincirinin bütün sonsal dağılımları, bu önsel
dağılımla çözümlenmektedir. R programlama dilinde yazılmış "BoomSpikeSlab"
paketi bütün sonsal dağılımların girilmiş olduğu bir paket olup çalışmada bu paketten
yararlanılmıştır.
Çalışmada nihai hedefi Yeni Zelanda'nın tektonik birlikleri hakkında, GPS zaman
serileri kullanılarak, yorumlar yapılmasıdır. Bu amaçla kümeleme analizi yapılmak
istenmiştir. Bu çalışmada kümeleme analizi için özgün bir metot önerilmektedir. Bu
metot, her bir tahmin ettirici değişkenin, yanıt değişkeninin Markov zincirindeki
muhtemel 30000 regresyon modeline dahil olma oranlarını k-Ortalama kümeleme
algoritmasında kullanılmasıdır. Burada k küme sayısını belirtmekte olup küme sayısını
belirlemek için Dirsek Metodu (Elbow Method for k-Means Clustering) olarak bilinen
bir metot kullanılmış, ideal küme sayısının 3 olduğu düşünülmüştür.
Kümeleme analizi sonucu, bu çalışmada uygulanan tekniklerin, Yeni Zelanda'nın
tektonik birlikleri ve yavaş depremler fenomeni ile ilgili anlamlı yorumlar
yapılabileceği görülmüştür.

Abstract:
In the world of statistics, many methods of data processing and understanding
stochastic processes have developed over the years. Linear regression analysis, which
is one of these methods, was chosen to be used in this study. Regression methods were
impossible to use on large data sets without computers in the past. Thanks to the
development of technology, these methods, which are difficult to use on big data, have
been introduced, and methods such as regression have been brought to be used on big
data. Regression models are the models used in artificial intelligence and machine
learning fields today. Regression is nothing more than fitting curves on data points of
dependent and independent variables. The problem here is which of the potential
curves best represents the dataset. Another problem is that it is difficult to select a
potential regression model with a large data set. In this study, the potential number of
regression models is 280 due to the size of the data set. This number is 1.5 times larger
than the diameter of the universe in kilometers. The problem of choosing the most
realistic model out of so many potential models can be solved by algorithms known as
Markov Chains, which are based on the distribution of conditional probabilities, put
forward by the Russian mathematician Markov. Markov says that the probability
revealed as a result of Bernoulli's independent experiments can also be revealed by
dependent experiments. Bernoulli proved with the experiments he designed that the
probability of the occurrence of an event whose probability is desired to be predicted
should have a value close to the result of a series of independent experiments. In other
words, the probability that a marble to be drawn from a box containing 1000 red and
white marbles will come out red, according to Bernoulli, will be close to the rate of
coming red as a result of a series of independent experiments, and as the number of
experiments is increased, the test rate will converge to the true value. Bernoulli's most
important argument in these experiments is that each attraction must be independent
of the previous one. At this point, Markov put forward his revolutionary idea that these
experiments could produce a result without being independent of each other. Markov
also designed very clever experiments for this idea. Markov claims that the desired
probability can be obtained as a result of a series of interdependent probability
experiments (chains of probability experiments). Even if we cannot observe it in the
universe, all events are more or less dependent on each other. Events that we will never
anticipate have an impact on us or on another event. Often these effects are small
enough to be ignored. An example of this is the famous idea of the "Butterfly Effect"
mentioned by the famous science writer James Gleick. In Geology, for example, an
earthquake is triggered by a very small effect right before it starts. This extraordinary
claim of the world of probability enables to establish connections between events
thanks to the developed methods. In this study, it is aimed to reveal the connection between the ground movements of the existing stations in the GPS (Global Positioning
System) time series data set from New Zealand.
For the study, a GPS ground motion dataset with 146 GPS stations, which was taken
from New Zealand's Institute of Geological and Nuclear Sciences (GNS Sciences),
between 2011-2021, with daily measurements at micrometer (10-6 m) level, was used.
Missing data points on this data set were filled with the Expectation Maximization
algorithm. In statistics, the Expectation maximization algorithm is an iterative search
method used to find the maximum likelihood or maximum after-effects estimates of
the parameters of statistical models that depend on unobservable hidden variables.
Thanks to this method, the missing data is filled in as close to the truth as possible and
the analysis continues. After removing the missing data points, the time series of all
stations in 3 measurement directions were plotted. It has been observed in the graphs
that the eastward time series contain more specific ground movements. Therefore, only
the data in the east direction was used in the continuation of the analysis. Because the
trend component present in the time series causes high correlation and complicates the
analysis, the time series were de-trended. In the study, linear regression was desired.
However, the problem cannot be solved by classical regression analysis as there are
2
80 possible regression models due to the large data set. For the diagnosis of the most
reasonable regression model, Variable Selection with Stochastic Search was made and
a computer Gibbs sampling algorithm based on Markov Chains was used for this. The
Markov chain is designed to consist of 30000 chains. A distribution model known as
"Spike-and-Slab Prior" has been proposed for the prior distribution of the coefficients
in the regression model of potential plausible predictor variables in the Markov chain,
which is also a concept of Bayesian statistics. All posterior distributions of the
generated Markov chain are analyzed by this prior distribution. "BoomSpikeSlab"
package written in R programming language is a package where all posterior
distributions are entered and this package is used in the study.
The ultimate goal of the study is to make comments on New Zealand's tectonic units
using GPS time series. For this purpose, cluster analysis was desired. In this study, a
unique method is proposed for cluster analysis. This method is the use of each
predictor variable, the rate of inclusion of the response variable in the possible 30000
regression model in the Markov chain, in the k-means clustering algorithm. This
inclusion rate determines how important the relevant predictor variable is for the
response variable in regression, whether this predictor needs to be present in the
regression. The K-means clustering algorithm creates clusters as many as the specified
number of clusters through the coordinates entered for each element to be clustered.
While doing this, it assigns coordinates for the center of random clusters as much as
the determined number of k clusters, and the elements to be clustered are clustered in
the form of their distance from this center. In the next step, the centers of these
determined clusters are recalculated and all the elements become members of the
cluster center closest to them. This process ends when the cluster center does not
change again, and this last situation becomes the result of the cluster analysis. Here, k
indicates the number of clusters, and a method known as the Elbow Method for kMeans Clustering was used to determine the number of clusters, and the ideal number
of clusters was thought to be 3. In the graph of the method made for the estimation of
the cluster number, the number of clusters was not clearly revealed as 3. However, k=3
would still be the most reasonable choice. The reason why the number of clusters does
not appear clearly is the diversity of the effects on the ground movements, the
transitions between the clusters are not very sharp. As a result of the cluster analysis, it was seen that the techniques applied in this study
can make meaningful interpretations about New Zealand's tectonic units and slow
earthquakes phenomenon. The resulting clusters overlapped with slow seismic zones
defined in the literature. As a result of the k-mean clustering analyzes performed by
increasing the number of k clusters, the tendency of newly formed clusters to be
correlated with slow earthquakes continued. In the analysis performed by shifting 5
days for a specifically selected slow earthquake with a shorter period (150-day data
length), it was observed that the clusters that existed before the slow earthquake started
lost their cluster quality when the earthquake started and continued. This shows that
slow earthquakes have an effect on the effects on the surface movements.