-90 -45 0 45 90
order
45 90
degree
u a b
Figure 4.1: u) Unclassified, a) essential and b) nonessential classes in sc format
u a b
1000 3000 5000 7000
Figure 4.2: Number of coefficients, 937, 6204 and 1136 in u) Unclassified, a) essential and b) nonessential classes, respectively.
4.3 results
The essential and nonessential classes represented in Figure 4.1 acts as the source for the training process and the goal of the classification is to decide the association of the unclassified coefficients.
Since the source dataset consists of two classes, therefore,k=3. The number ofkif set to odd helps to avoid the tie situation. Let us call the coefficients of the essential and nonessential classes as the training points and the unclassified coefficients as the candidate points. Then, the training points of 16 years grace data are stored in the memory and for each unclassified candidate coefficient, the distance between the candidate and the training points are measured. The training points are sorted on the bases of their distance from the candidate point. As the labels of the training points are also given therefor the candidate point will be classified as the majority class of the top three training points of the sorted list. The process is repeated for all the candidate points. At the end of the process, it is found that out of 937 unclassified points 286 points are close to the essential and 651 to the nonessential class, as shown in Figure 4.3. So the total number of essential and nonessential coefficients comes out to be 6490 and 1787, respectively, as given in Figure 4.4 and 4.5
These results bring us to the end of the first step of classifying the sh coefficients. The combin-ations of three independent algorithms help us to identify the two distinct classes in the dataset, i.e. essential and nonessential. The coefficients in essential class posses bigger place in the feature 27
-90 -45 0 45 90 order
45
90
degree
a b
Figure 4.3: Classification of unclassified coefficients given in Figure 4.1 into a) essential and b) nonessential classes in sc format
-90 -45 0 45 90
order 45
90
degree
a b
Figure 4.4: a) essential and b) nonessential classes in sc format
a b
1 3 5 7
×103
Figure 4.5: Number of coefficients, 6490 and 1787 in a) essential and b) nonessential classes, respectively.
space shows more yearly variance and therefore having the significant information of the gravity variation during the year. While the nonessential class is denser in feature space, therefore, possess a small place in the feature space, show less yearly variance and therefore having non-significant information of the gravity variation during the year. As the second step of machine learning, in the next chapter, another entirely different and independent machine learning technique, i.e. ann, has been described along with its development. In the forthcoming chapters, ann has been used as asupervised classifierandprediction tool.
artificial neural networks (ann) 5
A
computer is a machine which works on instructions. Now, the demand includes that the computers must learn from the observations as humans do. This is the basic concept of machine learning. Since its invention, the computer is analyzing the data, which are collected by humans or captured by the sensors, using the known theoretical or mathematical models. Now there is a need to look at the data from another viewpoint. Hence, rather than theoretical or mathematical models, the data-driven analysis is in vogue. Human expects that the computer must look at the data in a way the humans have not looked at it and unveil the facts and hidden behavior or the relationships among its different parameters and finally make optimal decisions.ann is one of the effective tools of machine learning. Web searching based on character, image or sound recognition, self driven vehicles using driver assistance systems, robots doing sorting, quality control, analyzing products and performing other importing jobs, face and object detection in cameras, text recognition form images and videos, online gaming, real time analysis of behavior, weather and natural hazard prediction, music composition, sound generation are some of its most common application fields. The salient feature of the ann is learning. It learns from the historical data and improves its ability to do decisions. This study attempts to use the ann for the data miningof the grace monthly datasets and uses it for two different purposes, i.e.
• classificationand
• predictionof grace sh coefficients.
The objective of the study is to analyze the behavior of the grace monthly sh coefficients and find patterns, trends, classes or groups among them. In this chapter, the discussion starts from the introduction of ann followed by its applications in the coming chapters for the classification and prediction of grace sh coefficients. The important steps towards the implementation of ann are,
1. source and target datasets 2. network architecture
3. activation function 4. evaluation criteria 5. feedforward 6. optimization
7. error backpropagation
ann is a data processing unit. The basic components of an ann are source dataset, target datasetand anetwork of neurons. Requirements determine the size of the ann, that is, the number of layers in the network and the number of neurons in each layer. The first is the input and the last is the output layer. Between the input and the output layers there exist inner layers. Except the neurons in the first layer, each neuron has a numerical value associated with it, named as bias, while the neurons in the first layer receive numerical values as input from the source dataset.
Each neuron in a layer is connected to each neuron of the next layer. The connection of a neuron to the other neuron also has a numerical value, named asweight. The neurons, except the input layer, containactivation functions. During the learning process weights and biases get new values, due to which the output values changes. Activation function suppresses too small changes and ensures that only the essential information is passed on. All applications require a trained, tested and validated ann. Source dataset is generally a large dataset and a big part∼60-70% of its trains, and two relatively small parts∼15-20% validates and∼15-20% tests the ann, for instance. Target dataset is the required output and training, which is an iterative process, tunes the weights and biases of the network to produce an output like target dataset. The following section explains the components in details.
5.1 source and target datasets
In ann, learning is a collection of three processes i.e. training, validation and test. During the training, an ann using source data matrix learns how to reach closer to the target value (Goodfel-low et al., 2016). In this chapterPwith orderκ×q denotes the source data matrix withκ is the number of features in the data andqare the data samples, ranging from 1 to q, whileTwith order m×q represents the target vector with m targets correspond to each sample in the source data matrix.
P = h
p1 p2 . . . pq . . . pq
i
κ×q , T = h
t1 t2 . . . tq . . . tq
i
m×q .
(5.1)
wherepq andtq are the input sample vectors and corresponding target vectors, respectively.
5.2. feedforward network architecture