Overview of Time Series Data Learning
Time-series data is called data whose values change over time, such as stock prices, temperatures, and traffic volumes. By applying machine learning to this time-series data, a large amount of data can be learned and used for business decision making and risk management by making predictions on unknown data.
Time-series data includes trends, seasonality, and random elements. Trends represent long-term trends, seasonality is cyclical patterns, and random elements are unpredictable noise. To take these factors into account, various methods are used in forecasting time-series data.
Among them, ARIMA described in “Examples of implementations for general time series analysis using R and Python”, Prophet described in “Time series analysis using Prophet“, LSTM described in “Overview of LSTM and Examples of Algorithms and Implementations“, and state-space models are representative methods used. These methods are prediction methods based on machine learning, which learns from past time-series data to predict the future.
Since time-series data includes time, in order to learn from past data and predict the future, it is necessary to process the time-series data appropriately, for example, by decomposing the data into trend, seasonal, and residual components.
Here, we discuss autocorrelation analysis, one of the basic methods of time-series data analysis.
Autocorrelation analysis is a method to analyze the correlation between past and current values of time-series data, and enables us to predict the next value by examining the correlation between past and current values.
The index in autocorrelation analysis is called the autocorrelation coefficient, which is calculated as the correlation between a value at a certain point in time-series data and a value at a certain point in the past. The autocorrelation coefficient takes values from -1 to 1, with values closer to 1 indicating a strong positive correlation, values closer to -1 indicating a strong negative correlation, and values closer to 0 indicating no correlation.
Since the autocorrelation coefficient varies depending on the lag of the time-series data (lag 1 represents the period from a past point in time to the current point in time; lag 1 refers to one previous point in time and lag 2 refers to two previous points in time), the autocorrelation coefficient is calculated with varying lags.
The following are the main methods used to analyze time series data using the autocorrelation coefficient.
- Autocorrelation graph: By graphing the autocorrelation coefficient against lags, patterns such as seasonality and periodicity of data can be visually grasped.
- Auto-regressive model (AR model): This model predicts the future from past data using information on autocorrelation coefficients. This method is effective when the autocorrelation coefficient is high.
- Moving average autoregressive model (ARMA model): This model combines the autoregressive model with a moving average model. Using the information from the autocorrelation and partial autocorrelation coefficients, it is possible to predict the future from past data.
- Autoregressive moving average model (ARIMA model): This model combines the ARMA model with a difference model. This model takes into account the non-stationarity of the data and is often used in the analysis of time series data.
Next, we will discuss the state-space model, which is a generalized model that further develops the idea of autocorrelation.
The state-space model is one of the statistical models often used in the analysis of time-series data and is a very general framework. In the state-space model, observed time-series data are considered to be generated by some stochastic process, and a mathematical model is constructed to describe the process. It also assumes a non-observable variable called a “state variable” that governs the stochastic process, and considers a transition model that represents the temporal change of the state variable and an observation model that generates the observed values from the state variable. Specifically, the state-space model has the following elements
- State equation: a model representing the temporal change of the state variable
- Observation equation: a model that generates the observed values from the state variable
- Initial state distribution: probability distribution of the initial state variable
- Error distribution: probability distribution assumed as the error term in the state equation and the observation equation
In time series data analysis using the state space model, the objective is to estimate the state variables for the observed data, and algorithms such as the Kalman filter and particle filter are used for this estimation. By using these algorithms, it is possible to calculate the estimated value of the state variable at the current point in time and the accuracy of that estimate.
Time series data analysis using state space models can also be used to predict future values using the estimated results of the state variables, and may be used for applications such as predicting future values from observed equations using the estimated values of the state variables, detecting anomalous values, and completing missing values.
Time series data analysis
A time series is a series of values (a series of values) obtained by continuously (or discontinuously at regular intervals) observing changes in a phenomenon over time. For example, it is a sequence of data measured over time in statistics or signal processing, and is measured at a (usually constant) time interval. If the interval is not uniform, it is called a point process.
Examples of time series and point process plots are shown below.
Recording an observation target as time series data implicitly assumes that the observation target is changing in continuous time. In fact, when we draw a time series chart, we interpolate between the points of observation (the time when no observation has been made) with the simplest function called a straight line to reflect the fact that the observation target is changing in continuous time. In other words, collecting time series data is a process of “assuming that the observation target or its attributes change in continuous time, and recording them at predetermined time intervals.
If we assume that the observed objects are changing in continuous time, one of the main purposes of time series analysis is to model the observed objects, i.e., to read out from the time series data a function whose argument is continuous time that represents the observed objects well. However, finding such a function is difficult because of the following points.
- The form of the function of continuous time is unknown. Moreover, since there are an infinite number of possible functions of continuous time, it becomes physically impossible to try to fit all of them.
- The form of the function that the observable follows may change over time. In other words, there is no guarantee that the function to be estimated will be the same throughout the period.
- Even if the form of the function is explicitly known (i.e., forced by the analyst), accurate estimation of the parameters of the function is not easy because of the noise in the observation.
In order to deal with these issues, the actual analysis of time series data aims to narrow down the class of functions that the observation target will follow through the knowledge of observers and analysts, previous studies, and analysis of preliminary data, and to represent the observation target using the simplest possible time series model among them.
The most common simplification used in time series analysis is to replace a continuous time function with a discrete time function. This means to convert continuous data like the one on the left into discrete data like the one on the right. This can be modeled by a probability distribution that a certain phenomenon will occur at each observation time.
The probability distribution to be assumed here is basically to be chosen by looking at the distribution of the target data, but if it is unclear, it is better to start thinking from the normal distribution that is used in many models.
In this blog, I will discuss the following about this time series data analysis.
Implementation of time series data analysis
Time-series data is called data whose values change over time, such as stock prices, temperatures, and traffic volumes. By applying machine learning to this time series data, a large amount of data can be learned and used for business decision making and risk management by making predictions on unknown data. This section describes the implementation of time series data using python and R.
Prophet is a time series forecasting tool developed by Facebook that can forecast future time series data, taking into account the effects of time flow, periodicity, holidays, etc. Prophet can be used in various fields such as business, finance, climate, and medicine. We describe a Python-based implementation of Prophet here.
Exponential Smoothing is a statistical method used for forecasting and smoothing time series data, especially for forecasting future values based on past observations. Exponential smoothing is a simple but effective method that allows for weighting against time and adjusting for the effect of past data.
- Overview of Kalman Filter Smoother and Examples of Algorithms and Implementations
Kalman Filter Smoother, a type of Kalman filtering, is a technique used to improve state estimation of time series data. The method usually models the state of a dynamic system and combines it with observed data for more precise state estimation.
Time-series data is called data whose values change over time, such as stock prices, temperatures, and traffic volumes. By applying machine learning to this time-series data, a large amount of data can be learned and used for business decision making and risk management by making predictions on unknown data. In this article, we will focus on state-space models among these approaches.
Dynamic Graph Embedding is a technique for analyzing time-varying graph data, such as dynamic networks and time-varying graphs. While conventional embedding for static graphs focuses on obtaining a fixed representation of nodes, the goal of dynamic graph embedding is to obtain a representation that corresponds to temporal changes in the graph.
Spatio-Temporal Graph Convolutional Network (STGCN) is a convolution for time-series data on a graph consisting of nodes and edges. Recurrent Neural Network, RNN), which is a model used to predict time variation instead of a recurrent neural network (RNN). This is an effective approach for data where geographic location and temporal changes are important, such as traffic flow and weather data.
- Dynamic Linear Model (DLM) Overview, Algorithm and Implementation Example
A Dynamic Linear Model (DLM) is a form of statistical modeling that accounts for temporal variation, and this model will be used to analyze time-series data and time-dependent data. Dynamic linear models are also referred to as linear state-space models.
- Overview of Dynamic Bayesian Networks (DBN) and Examples of Algorithms and Implementations
Dynamic Bayesian Network (DBN) is a type of Bayesian Network (BN), which is a type of probabilistic graphical model used for modeling time-varying and serial data. DBN is a powerful tool for time series and dynamic data and has been applied in various fields.
Dynamic Graph Neural Networks (D-GNN) are a type of Graph Neural Networks (GNN) designed to deal with dynamic graph data, where nodes and edges change with time. It is designed to handle data in which nodes and edges change over time. (For more information on GNNs, see “Graph Neural Networks: Overview, Applications, and Example Python Implementations. The approach has been used in a variety of domains including time series data, social network data, traffic network data, and biological network data.
Tensor decomposition (TD) is a method for approximating high-dimensional tensor data to low-rank tensors. This technique is used for data dimensionality reduction and feature extraction and is a useful approach in a variety of machine learning and data analysis applications. The application of tensor decomposition methods to dynamic module detection is relevant to tasks such as time series data and dynamic data module detection.
- ST-GCN (Spatio-Temporal Graph Convolutional Networks) Overview, Algorithm and Examples of Implementation
ST-GCNs (Spatio-Temporal Graph Convolutional Networks) are a type of graph convolutional networks designed to handle video and temporal data. data), this method can perform feature extraction and classification by considering both spatial information (relationships between nodes in the graph) and temporal information (consecutive frames or time steps). It is primarily used for tasks such as video classification, motion recognition, and sports analysis.
- Overview of DynamicTriad and Examples of Algorithms and Implementations
DynamicTriad is a method for modeling temporal changes in dynamic graph data and predicting node correspondences. This approach has been applied to predicting correspondences in dynamic networks and understanding temporal changes in nodes.
Techniques for analyzing graph data that changes over time have been applied to a variety of applications, including social network analysis, web traffic analysis, bioinformatics, financial network modeling, and transportation system analysis. Here we provide an overview of this technique, its algorithms, and examples of implementations.
Snapshot Analysis (Snapshot Analysis) is a method of data analysis that takes into account changes over time by using snapshots of data at different time points (instantaneous data snapshots). This approach helps analyze data sets with information about time to understand temporal patterns, trends, and changes in that data, and when combined with Graphical Data Analysis, allows for a deeper understanding of temporal changes in network and relational data. This section provides an overview of this approach and examples of algorithms and implementations.
Dynamic Community Detection (Dynamic Community Analysis) will be a technique for tracking and analyzing temporal changes in communities (modules or clusters) within a network with time-relevant information (dynamic network). Usually targeting graph data (dynamic graphs) whose nodes and edges have time-related information, the method has been applied in various fields, e.g., social network analysis, bioinformatics, Internet traffic monitoring, financial network analysis, etc. It is used in the following areas.
Dynamic Centrality Metrics is a type of graph data analysis that takes into account changes over time. Usual centrality metrics (e.g., degree centrality, mediation centrality, eigenvector centrality, etc.) are suitable for static networks and It provides a single snapshot of the importance of a node. However, since real networks often have time-related elements, it is important to consider temporal changes in the network.
Dynamic module detection is a method of graph data analysis that takes time variation into account. This method tracks changes in communities (modules) in a dynamic network and identifies the community structure at different time snapshots. Here we present more information about dynamic module detection and an example implementation.
Dynamic Graph Embedding is a powerful technique for graph data analysis that takes temporal variation into account. This approach aims to have a representation of nodes and edges on a time axis when graph data varies along time.
Network alignment is a technique for finding similarities between different networks or graphs and mapping them together. By applying network alignment to graph data analysis that takes into account temporal changes, it is possible to map graphs of different time snapshots and understand their changes.
Graph data analysis that takes into account changes over time using a time prediction model is used to understand temporal patterns, trends, and predictions in graphical data. This section discusses this approach in more detail.
TIME-SI (Time-aware Structural Identity) is one of the algorithms or methods for identifying structural correspondences between nodes in a network, taking into account time-related information. It will be used in a variety of network data, including social networks.
Displaying and animating graph snapshots on a timeline is an important technique for analyzing graph data, as it helps visualize changes over time and understand the dynamic characteristics of graph data. This section describes libraries and implementation examples used for these purposes.
This paper describes the creation of animations of graphs by combining NetworkX and Matplotlib, a technique for visually representing dynamic changes in networks in Python.
Methods for plotting high-dimensional data in low dimensions using dimensionality reduction techniques to facilitate visualization are useful for many data analysis tasks, such as data understanding, clustering, anomaly detection, and feature selection. This section describes the major dimensionality reduction techniques and their methods.
Gephi is an open-source graph visualization software that is particularly suitable for network analysis and visualization of complex data sets. Here we describe the basic steps and functionality for visualizing data using Gephi.
Cytoscape.js is a graph theory library written in JavaScript that is widely used for visualizing network and graph data. Cytoscape.js makes it possible to add graph and network data visualization to web and desktop applications. Here are the basic steps and example code for data visualization using Cytoscape.js.
- Visualization of Graph Data Using Sigma.js
Sigma.js is a web-based graph visualization library that can be a useful tool for creating interactive network diagrams. Here we describe the basic steps and functions for visualizing graph data using Sigma.js.
Automatic machine learning (AutoML) refers to methods and tools for automating the process of designing, training, and optimizing machine learning models.AutoML is particularly useful for users with limited machine learning expertise or those seeking to develop efficient models, with the following main goals. This section provides an overview of this AutoML and examples of various implementations.
The Dynamic Factor Model (DFM) is one of the statistical models used in the analysis of multivariate time series data, which explains the variation of data by decomposing multiple time series variables into common factors (factors) and individual factors (specific factors). This is a model that explains data variation by decomposing multiple time series variables into common factors and individual factors (specific factors). This paper describes various algorithms and applications of DFM, as well as their implementations in R and Python.
Similarity is a concept that describes the degree to which two or more objects or things have common features or properties and are considered similar to each other, and plays an important role in evaluating, classifying, and grouping objects in terms of comparison and relatedness. This section describes the concept of similarity and general calculation methods for various cases.
Bayesian Structural Time Series Model (BSTS) is a type of statistical model that models phenomena that change over time and is used for forecasting and causal inference. This section provides an overview of BSTS and its various applications and implementations.
- Overview of Vector Autoregression Models and Examples of Applications and Implementations
Vector Autoregression Model (VAR model) is one of the time series data modeling methods used in fields such as statistics and economics, etc. VAR model is a model that is applied when multiple variables interact with each other. The general autoregression model (Autoregression Model) expresses the value of a variable as a linear combination of its past values, and the VAR model extends this idea to multiple variables, becoming a model that predicts current values using past values of multiple variables.
Online learning is a method of learning by sequentially updating a model in a situation where data arrives sequentially. Unlike batch learning in ordinary machine learning, this algorithm is characterized by the fact that the model is updated each time new data arrives. This section describes various algorithms and examples of applications of on-run learning, as well as examples of implementations in python.
LightGBM is a Gradient Boosting Machine (GBM) framework developed by Microsoft, which is a machine learning tool designed to build fast and accurate models for large data sets. Here we describe its implementation in pyhton, R, and Clojure.
Robust Principal Component Analysis (RPCA) is a method for finding a basis in data, and is characterized by its robustness to data containing outliers and noise. This paper describes various applications of RPCA and its concrete implementation using pyhton.
This section provides an overview of python Keras and examples of its application to basic deep learning tasks (handwriting recognition using MINIST, Autoencoder, CNN, RNN, LSTM).
RNN (Recurrent Neural Network) is a type of neural network for modeling time-series and sequence data, and can retain past information and combine it with new information, such as speech recognition, natural language processing, video analysis, and time series prediction, It is a widely used approach for a variety of tasks.
LSTM (Long Short-Term Memory) is a type of recurrent neural network (RNN), which is a very effective deep learning model mainly for time series data and natural language processing (NLP) tasks. LSTM can retain historical information and model long-term dependencies, making it a suitable method for learning long-term information as well as short-term information.
- About GRU (Gated Recurrent Unit)
GRU (Gated Recurrent Unit) is a type of recurrent neural network (RNN) that is widely used in deep learning models, especially for processing time series data and sequence data. The GRU is designed to model long-term dependencies in the same way as the LSTM (Long Short-Term Memory) described in “Overview of LSTM and Examples of Algorithms and Implementations,” but it is characterized by its lower computational cost than the LSTM. It is characterized by lower computational cost than LSTM.
- About Bidirectional RNN (BRNN)
Bidirectional Recurrent Neural Network (BRNN) is a type of recurrent neural network (RNN) model that can simultaneously consider past and future information. BRNN is particularly useful for processing sequence data and is widely used in tasks such as natural language processing and It is widely used in tasks such as natural language processing and speech recognition.
- About Deep RNN
Deep RNN (Deep Recurrent Neural Network) is a type of recurrent neural network (RNN), which is a stacked model of multiple RNN layers. deep RNN helps model complex relationships in sequence data and extract more sophisticated feature representations. Typically, a Deep RNN consists of RNN layers stacked in multiple layers in the temporal direction.
- About Stacked RNN
Stacked RNN (Stacked Recurrent Neural Network) is a type of recurrent neural network (RNN) architecture that uses multiple RNN layers stacked on top of each other, enabling modeling of more complex sequence data and effectively capturing long-term dependencies It is a method that allows for more complex sequence data modeling and the ability to effectively capture long-term dependencies.
- About Echo State Network (ESN)
Echo State Network (ESN) is a type of reservoir computing, a type of recurrent neural network (RNN) used for prediction, analysis, and pattern recognition of time series and sequence data. tasks and may perform well in a variety of tasks.
- Overview of Pointer-Generator Networks, Algorithms, and Examples of Implementations
The Pointer-Generator network is a type of deep learning model used in natural language processing (NLP) tasks, and is particularly suited for tasks such as abstract sentence generation, summarization, and information extraction from documents. The network is characterized by its ability to copy portions of text from the original document verbatim when generating sentences.
Elasticsearch is an open source distributed search engine for search, analysis, and data visualization that also integrates Machine Learning (ML) technology and can be leveraged for data-driven insights and predictions. It is a platform that can be used to achieve data-driven insights and predictions. This section describes various uses and specific implementations of machine learning technology in Elasticsearch.
In this article, we will discuss time series data. A time series is a series of data consisting of regularly observed values of a certain quantity arranged according to their measurement time. In order to predict future values of a time series, it is necessary that future values are based to some extent on past values. In this article, we will discuss the implementation of AR, MA, and ARMA models using Clojure.
In this article, we describe an implementation of the Kalman filter, one of the applications of the state-space model, in Clojure. The Kalman filter is an infinite impulse response filter used to estimate time-varying quantities (e.g., position and velocity of an object) from discrete observations with errors, and is used in a wide range of engineering fields such as radar and computer vision due to its ease of use. Specific examples of its use include integrating information with errors from device built-in accelerometers and GPS to estimate the ever-changing position of vehicles, as well as in satellite and rocket control.
The Kalman filter is a state-space model with hidden states and observation data generated from them, similar to the hidden Markov model described previously, in which the states are continuous and changes in state variables are statistically described using noise following a Gaussian distribution.
In this article, we will discuss structural change estimation of time series data as an application of nonparametric Bayesian models. One of the problems in the analysis of time series data is the estimation of changes in the structure of the data. The problem of analyzing changes in the properties of data is an important topic that has been extensively studied as change checking. Here, we describe a method using a statistical model such as the Dirichlet process.
The basic idea is to assume that each data is generated from multiple models with a certain probability, and to estimate structural changes in the data by estimating changes in the generation process over time.
There are various extensions and generalizations of the Bandit problem in addition to the linear Bandit. In settings such as news recommendation, the probability distribution of a quantity corresponding to the reward of the bandit problem, such as the presence or absence of clicks, may vary with time. There are several possible formulations of such settings, and various measures are possible depending on the formulation. Here we discuss some of the most representative of them.
- Introduction to Customer Flow Line Research HMM, k-medoids, fractal dimension, kernel density estimation method
- Time series forecasting with GRU
- Change detection using subspace method -Singular spectral transform method for time series data
- Prior probability model with Stan and dlm (external link)
- How to train multiple time-series data with a single deep learning model using Keras (external link)
There are two methods for training multiple time-series data with a single deep learning model in Keras. (The advantage of the first method is that the model is simpler and therefore faster to learn and predict than the second method, while the advantage of the second method is that it can be customized for each time series, making it easier to improve accuracy than the first method.
- I used Prophet, a time series analysis library (external link)Prophet is a time series analysis algorithm developed by Meta, the company behind Facebook. Since the company released Prophet software (there are libraries for Python and R) as open source software (OSS) in 2017, it has rapidly spread in the field of time series forecasting. It is now so widespread that it can be said that “Prophet is basically used for time series analysis of daily data (data recorded on a daily basis).
Theory of Time Series Data Analysis
- Analysis of Time Series Data (1) Introduction
- Analysis of Time Series Data (2) Basics of statistical analysis
- Analysis of Time Series Data (3) Analysis of Time Dependent Data and AR Model
- Analysis of Time Series Data(4) ARCH Model and GARCH Model
- Sparsity of Time Transitions
Application of state-space models to time-series data analysis
A state-space model is a model for dealing with many time series models in a unified manner, and provides a framework for dealing with many problems related to time series, such as forecasting, storage, component analysis, and parameter estimation, in a unified manner as a problem of state estimation. In this section, we describe what a state-space model is.
The specific state space models we will discuss are linear and Gaussian state space models, AR models, autoregressive and moving average (ARMA) models, component decomposition models, and time-varying coefficient models.
- Time series data analysis (2) Filtering Sequential estimation of state and seasonal adjustment model
Given a time series of observed values Yj={y1,…,yn} and a state space model, the problem of estimating the state sn is called state estimation. Depending on the relationship between the last time point n of the observed values and the time j of the state to be estimated, the problem can be divided into three cases: smoothing, filtering, and forecasting.
The reason for considering the state estimation problem in the state-space model is that most of the problems required to solve real-world problems, such as time series prediction, interpolation, parameter estimation, and component decomposition, can be solved in a unified manner by using this state estimation method. Aside from online control and sequential forecasting, which require real-time processing, smoothing generally uses information before and after the point of interest in information extraction and estimation, so it can provide more accurate estimation than forecasting or filtering.
For linear and Gaussian state-space models, the conditional distribution of states p(xn|Yn) is a normal distribution and can be efficiently computed by the Kalman filter. On the other hand, if the model is nonlinear or the noise distribution is non-Gaussian, the predictive distribution of the states will be non-Gaussian, and it is necessary to approximate it in some way.In this article, we will mainly discuss the approach using the particle filter.
The most commonly used packages for handling state-space models in R are dlm and KFAS. Both of them allow filtering, smoothing, and prediction using the Kalman filter, but they differ in some respects. Here, we will discuss the analysis of state-space models using the dlm package, which is a package developed by Giovanni Perris. The dlm package is developed by Giovanni Perris and handles the dynamic liner model, which is a linear and normally distributed state-space model.
Since the data analyzed in the previous dlm analysis clearly shows seasonal variations, we will try to add seasonal adjustment to the model. dlmModSeas or dlmModTrig functions are used to handle the seasonal adjustment component in dlm. The former uses dummy variables to represent the seasonal adjustment component, while the latter uses trigonometric functions. Here we used the dlmModSeas function.
KFAS is a package developed by Jouni Heleke that differs from dlm in that it has a coefficient matrix called Rt over system noise, which is used to select which states to add system noise to.
KFAS also analyzes the same seasonal adjustment model as described above. in KFAS, the model is defined by the SSModel function. in KFAS, as in dlm, the model is built by combining functions. In the code below, the model is constructed by combining SSMtrend, a function that handles multinomial components, and SSMseasonal, a function that handles seasonal adjustment components. The degree argument is given 1 to make it a local-level model.
Create a “particle filter” in R without using any packages. If it is just filtering, it can be written in almost 3 lines, except for the initialization and parameter setting parts. Basically, for the data (observed values), the closer the particles are to the observation at time t, the greater the weight, the more likely they are to be selected in the resampling step, and thus the closer the path is to the data. The distribution of particles at each time point then represents the posterior distribution (more precisely, the filtered distribution) obtained from the model.
The field of marketing has long neglected “dynamic structural understanding. One of the reasons for this is that data that could withstand dynamic analysis did not exist. However, a larger problem must be pointed out, which is a kind of lack of appreciation for the idea of “dynamic. Many marketing researchers believed that it was sufficient to conduct analysis based on the assumption of a “static consumer image.
There is no validity in the idea that consumer attitudes and behavior do not change dynamically, and unless we consider them dynamic and model them to gain knowledge, we will not be able to gain a deeper understanding of the demand side. Recently, this situation has changed, and the importance of explicitly considering time to understand marketing phenomena has been understood. The accumulation of big data-type consumer behavior data (people x products/services x time points) has progressed, and dynamic analysis with an active awareness of “time points” is now being conducted.
The question of whether there is a causal relationship from one time series to another assumes that at least two time series data are of interest. Causal inference based on time series is essentially a multivariate time series problem, and multivariate autoregression models (Vector AutoRegression model, VAR model) are often used as models.
Here, we describe the procedure for analyzing causality based on the VAR model, using the free software R as an example of the causal relationship between the approval rating of the Cabinet and stock prices. Data often contain missing data, and the Cabinet approval rating, which is the subject of this paper, is no exception. In this section, we will discuss interpolation of deficient values using the function decomp included in the timsac package of R. In addition, since causal analysis using the VAR model assumes stationarity of time series, it is necessary to check for stationarity and non-stationarity and to perform preprocessing to make the time series stationary. The procedure for this and the use of the unit root test are also described. The R package vars is used for estimation of uncontrolled/controlled VAR models, lag selection, causality tests, and calculation of impulse response functions.
In this section, we introduce the multivariate autoregressive model (VAR model) as a framework for analyzing the causality of time series. vars. A time series xt to yt is said to be “causal in the Granger sense” when the past values of other time series xt are useful in predicting time series yt.
The difference between a hidden Markov model (HMM) and a state space model is that in a hidden Markov model, the variables representing the “state” take a finite number of discrete values, whereas in a state space model they are continuous blood (vector values of real-valued components).
The forward algorithm of the hidden Markov model corresponds to the sequential filter (one-period-ahead prediction and filtering) of the state-space model, while the forward-backward algorithm corresponds to the smoothing formula. Note, however, that in the latter case, there is an apparent difference in the formulas often used.
When I hear that the age of the universe is 13.8 billion years old, I feel that the time scale of the universe and celestial bodies is incredibly long. Therefore, no one would expect that the sun, which set in the western sky yesterday evening, would come up a million times brighter in the morning today. But in the real universe, aside from ordinary stars like the sun, we frequently witness such astronomical observations that change on a human time scale.
The first reported explosion of V455 Andromeda was on September 4, 2007. The possibility of an explosion of this star had been pointed out for some time. Hiroshima University’s 1.5-meter telescope “Kanata” was pointed at this object in order to observe the oscillation phenomenon with a period of about 80 minutes, which is only observed in the early stages of the explosion due to its close distance from the earth. The brightness oscillation was successfully detected. The data obtained, however, showed that the temperature of the object becomes lower when it is brighter, which is unusual for a celestial body to exhibit such fluctuations. However, the results were as expected for this phenomenon.
Here we describe the results of tomographic reconstruction of the accretion disc shape based on the data obtained from these observations.
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