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Stat 510: applied time series analysis.

•   Overview
•   Materials
•   Assessment Plan
•   Prerequisites
•   Online Notes

Time series data are intriguing yet complicated information to work with. While this course will provide students with a basic understanding of the nature and basic processes used to analyze such data, you will quickly realize that this is a small first step in being able to confidently understand what trends might exist within a set of data and the complexities of being able to use this information to make predictions or forecasts. Yet, whether it is financial, medical or weather related, this type of data is quite frequently found in much of our daily lives.

Course Topics

Topics typically covered in this graduate level course include:

• Understanding the characteristics of time series data
• Understanding moving average models and partial autocorrelation as foundations for analysis of time series data
• Exploratory Data Analysis - Trends in time series data
• Using smoothing and removing trends when working with time series data
• Understanding how periodograms are used with time series data
• Implementing ARMA and ARIMA time series models
• Identifying and interpreting various patterns for intervention effects
• Examining the analysis of repeated measures design
• Using ARCH and AR models in multivariate time series contexts
• Using spectral density estimation and spectral analysis
• Using fractional differencing and threshold models with time series data
•   Statistical Modeling
•   Modeling with R
•   Time Series Methods

Course Author(s)

Dr. Megan Romer is the current author of the materials used in this course. The material builds on that of the course's previous authors, Robert Heckard and John Fricks.

This course makes extensive use of the R Statistical Software. This is open-source free software that can be downloaded from the R Project home page. For more information and links to download this software please see the  Statistical Software  page. MS Word is also required.

R involves programming. Students should be a quick learner of software packages. Students who have no experience with programming or are anxious about being able to manipulate software code are strongly encouraged to take the one-credit course in R in order to establish this foundation.  R will be supported and sample programs will be supplied but you will be required to do some programing on your own. Due to different software applications, software versions, and platforms, there may be issues with running code. Students must be proactive in seeking advice and help from appropriate sources, including documentation resources, the class discussion forum, the teaching assistant, instructor or helpdesk.

Shumway R.H., Stoffer, D.S. (2012). Time Series Analysis and Its Applications With R Examples , 4th Edition, Springer. ISBN: 978-3319524511

(The text is required, though students do not have to purchase it because it is available electronically through the Penn State library.)

Last updated: FA23

Assessment Plan

Lab / Homework Activities  - will be given weekly.  In order to receive credit for homework, all assignments must include HOW an answer is obtained, not just the numerical solution. These assignments can be compiled in Word, however, submission as a .pdf is prefered.

Exams  - There will be one mid-term and one final exam. These are 'take-home' application oriented exams that should be completed in the time specified by the instructor.

Prerequisites

STAT 462 - Applied Regression Analysis, or STAT 501 - Regression Methods, or STAT 511 - Regression Analysis and Modeling

Time Series Analysis Explained

Time series analysis is a powerful statistical method that examines data points collected at regular intervals to uncover underlying patterns and trends. This technique is highly relevant across various industries, as it enables informed decision making and accurate forecasting based on historical data. By understanding the past and predicting the future, time series analysis plays a crucial role in fields such as finance, health care, energy, supply chain management, weather forecasting, marketing, and beyond. In this guide, we will dive into the details of what time series analysis is, why it’s used, the value it creates, how it’s structured, and the important base concepts to learn in order to understand the practice of using time series in your data analytics practice. 

Table of Contents

• What Is Time Series Analysis? 
• Why Do Organizations Use Time Series Analysis? 
• Components of Time Series Data 

Types of Data

• Important Time Series Terms and Concepts

Time Series Analysis Techniques

• Advantages of Time Series Analysis
• Challenges of Time Series Analysis 
• The Future of Time Series Analysis.

What Is Time Series Analysis?

Time series analysis is indispensable in data science, statistics, and analytics. 

At its core, time series analysis focuses on studying and interpreting a sequence of data points recorded or collected at consistent time intervals. Unlike cross-sectional data, which captures a snapshot in time, time series data is fundamentally dynamic, evolving over chronological sequences both short and extremely long. This type of analysis is pivotal in uncovering underlying structures within the data, such as trends, cycles, and seasonal variations.

Technically, time series analysis seeks to model the inherent structures within the data, accounting for phenomena like autocorrelation, seasonal patterns, and trends. The order of data points is crucial; rearranging them could lose meaningful insights or distort interpretations. Furthermore, time series analysis often requires a substantial dataset to maintain the statistical significance of the findings. This enables analysts to filter out 'noise,' ensuring that observed patterns are not mere outliers but statistically significant trends or cycles.

To delve deeper into the subject, you must distinguish between time-series data, time-series forecasting, and time-series analysis. Time-series data refers to the raw sequence of observations indexed in time order. On the other hand, time-series forecasting uses historical data to make future projections, often employing statistical models like ARIMA (AutoRegressive Integrated Moving Average). But Time series analysis, the overarching practice, systematically studies this data to identify and model its internal structures, including seasonality, trends, and cycles. What sets time series apart is its time-dependent nature, the requirement for a sufficiently large sample size for accurate analysis, and its unique capacity to highlight cause-effect relationships that evolve.

Why Do Organizations Use Time Series Analysis?

Time series analysis has become a crucial tool for companies looking to make better decisions based on data. By studying patterns over time, organizations can understand past performance and predict future outcomes in a relevant and actionable way. Time series helps turn raw data into insights companies can use to improve performance and track historical outcomes.

For example, retailers might look at seasonal sales patterns to adapt their inventory and marketing. Energy companies could use consumption trends to optimize their production schedule. The applications even extend to detecting anomalies—like a sudden drop in website traffic—that reveal deeper issues or opportunities. Financial firms use it to respond to stock market shifts instantly. And health care systems need it to assess patient risk in the moment. 

Rather than a series of stats, time series helps tell a story about evolving business conditions over time. It's a dynamic perspective that allows companies to plan proactively, detect issues early, and capitalize on emerging opportunities.

Components of Time Series Data

Time series data is generally comprised of different components that characterize the patterns and behavior of the data over time. By analyzing these components, we can better understand the dynamics of the time series and create more accurate models. Four main elements make up a time series dataset:

• Seasonality

Trends show the general direction of the data, and whether it is increasing, decreasing, or remaining stationary over an extended period of time. Trends indicate the long-term movement in the data and can reveal overall growth or decline. For example, e-commerce sales may show an upward trend over the last five years.

Seasonality refers to predictable patterns that recur regularly, like yearly retail spikes during the holiday season. Seasonal components exhibit fluctuations fixed in timing, direction, and magnitude. For instance, electricity usage may surge every summer as people turn on their air conditioners.

Cycles demonstrate fluctuations that do not have a fixed period, such as economic expansions and recessions. These longer-term patterns last longer than a year and do not have consistent amplitudes or durations. Business cycles that oscillate between growth and decline are an example.

Finally, noise encompasses the residual variability in the data that the other components cannot explain. Noise includes unpredictable, erratic deviations after accounting for trends, seasonality, and cycles.

In summary, the key components of time series data are:

• Trends: Long-term increases, decreases, or stationary movement
• Seasonality: Predictable patterns at fixed intervals
• Cycles: Fluctuations without a consistent period
• Noise: Residual unexplained variability

Understanding how these elements interact allows for deeper insight into the dynamics of time series data.

When embarking on time series analysis, the first step is often understanding the type of data you're working with. This categorization primarily falls into three distinct types: Time Series Data, Cross-Sectional Data, and Pooled Data. Each type has unique features that guide the subsequent analysis and modeling.

• Time Series Data: Comprises observations collected at different time intervals. It's geared towards analyzing trends, cycles, and other temporal patterns. ‍ ‍
• Cross-Sectional Data: Involves data points collected at a single moment in time. Useful for understanding relationships or comparisons between different entities or categories at that specific point. ‍ ‍
• Pooled Data: A combination of Time Series and Cross-Sectional data. This hybrid enriches the dataset, allowing for more nuanced and comprehensive analyses.

Understanding these data types is crucial for appropriately tailoring your analytical approach, as each comes with its own set of assumptions and potential limitations.

Important Time Series Terms & Concepts

Time series analysis is a specialized branch of statistics focused on studying data points collected or recorded sequentially over time. It incorporates various techniques and methodologies to identify patterns, forecast future data points, and make informed decisions based on temporal relationships among variables. This form of analysis employs an array of terms and concepts that help in the dissection and interpretation of time-dependent data.

• Dependence : The relationship between two observations of the same variable at different periods is crucial for understanding temporal associations. ‍
• Stationarity : A property where the statistical characteristics like mean and variance are constant over time; often a prerequisite for various statistical models. ‍
• Differencing : A transformation technique to turn stationary into non-stationary time series data by subtracting consecutive or lagged values. ‍
• Specification : The process of choosing an appropriate analytical model for time series analysis could involve selection criteria, such as the type of curve or the degree of differencing. ‍
• Exponential Smoothing : A forecasting method that uses a weighted average of past observations, prioritizing more recent data points for making short-term predictions. ‍
• Curve Fitting : The use of mathematical functions to best fit a set of data points, often employed for non-linear relationships in the data. ‍
• ARIMA (Auto Regressive Integrated Moving Average) : A widely-used statistical model for analyzing and forecasting time series data, encompassing aspects like auto-regression, integration (differencing), and moving average.

Time series analysis is critical for businesses to predict future outcomes, assess past performances, or identify underlying patterns and trends in various metrics. Time series analysis can offer valuable insights into stock prices, sales figures, customer behavior, and other time-dependent variables. By leveraging these techniques, businesses can make informed decisions, optimize operations, and enhance long-term strategies.

Time series analysis offers a multitude of benefits to businesses.The applications are also wide-ranging, whether it's in forecasting sales to manage inventory better, identifying the seasonality in consumer behavior to plan marketing campaigns, or even analyzing financial markets for investment strategies. Different techniques serve distinct purposes and offer varied granularity and accuracy, making it vital for businesses to understand the methods that best suit their specific needs.

• Moving Average : Useful for smoothing out long-term trends. It is ideal for removing noise and identifying the general direction in which values are moving.
• Exponential Smoothing : Suited for univariate data with a systematic trend or seasonal component. Assigns higher weight to recent observations, allowing for more dynamic adjustments.
• Autoregression : Leverages past observations as inputs for a regression equation to predict future values. It is good for short-term forecasting when past data is a good indicator.
• Decomposition : This breaks down a time series into its core components—trend, seasonality, and residuals—to enhance the understanding and forecast accuracy.
• Time Series Clustering : Unsupervised method to categorize data points based on similarity, aiding in identifying archetypes or trends in sequential data.
• Wavelet Analysis : Effective for analyzing non-stationary time series data. It helps in identifying patterns across various scales or resolutions.
• Intervention Analysis : Assesses the impact of external events on a time series, such as the effect of a policy change or a marketing campaign.
• Box-Jenkins ARIMA models : Focuses on using past behavior and errors to model time series data. Assumes data can be characterized by a linear function of its past values.
• Box-Jenkins Multivariate models : Similar to ARIMA, but accounts for multiple variables. Useful when other variables influence one time series.
• Holt-Winters Exponential Smoothing : Best for data with a distinct trend and seasonality. Incorporates weighted averages and builds upon the equations for exponential smoothing.

The Advantages of Time Series Analysis

Time series analysis is a powerful tool for data analysts that offers a variety of advantages for both businesses and researchers. Its strengths include:

• Data Cleansing : Time series analysis techniques such as smoothing and seasonality adjustments help remove noise and outliers, making the data more reliable and interpretable. ‍
• Understanding Data : Models like ARIMA or exponential smoothing provide insight into the data's underlying structure. Autocorrelations and stationarity measures can help understand the data's true nature. ‍
• Forecasting : One of the primary uses of time series analysis is to predict future values based on historical data. Forecasting is invaluable for business planning, stock market analysis, and other applications. ‍
• Identifying Trends and Seasonality : Time series analysis can uncover underlying patterns, trends, and seasonality in data that might not be apparent through simple observation. ‍
• Visualizations : Through time series decomposition and other techniques, it's possible to create meaningful visualizations that clearly show trends, cycles, and irregularities in the data. ‍
• Efficiency : With time series analysis, less data can sometimes be more. Focusing on critical metrics and periods can often derive valuable insights without getting bogged down in overly complex models or datasets. ‍
• Risk Assessment : Volatility and other risk factors can be modeled over time, aiding financial and operational decision-making processes.

Challenges of Time Series Analysis

While time series analysis has a lot to offer, it also comes with its own set of limitations and challenges, such as:

• Limited Scope : Time series analysis is restricted to time-dependent data. It's not suitable for cross-sectional or purely categorical data.
• Noise Introduction : Techniques like differencing can introduce additional noise into the data, which may obscure fundamental patterns or trends.
• Interpretation Challenges : Some transformed or differenced values may need more intuitive meaning, making it easier to understand the real-world implications of the results.
• Generalization Issues : Results may only sometimes be generalizable, primarily when the analysis is based on a single, isolated dataset or period.
• Model Complexity : The choice of model can greatly influence the results, and selecting an inappropriate model can lead to unreliable or misleading conclusions.
• Non-Independence of Data : Unlike other types of statistical analysis, time series data points are not always independent, which can introduce bias or error in the analysis.
• Data Availability : Time series analysis often requires many data points for reliable results, and such data may not always be easily accessible or available.

The Future of Time Series Analysis

The future of time series analysis will likely see significant advances thanks to innovations in machine learning and artificial intelligence. These technologies will enable more sophisticated and accurate forecasting models while also improving how we handle real-world complexities like missing data and sparse datasets.

Some key developments are likely to include:

• Hybrid models strategically combine multiple techniques —such as ARIMA, exponential smoothing, deep learning LSTM networks, and Fourier transforms—to capitalize on their respective strengths. Blending approaches in this way can produce more robust and precise forecasts.
• Advanced deep learning algorithms like LSTM recurrent neural networks can uncover subtle patterns and interdependencies in time series data. LSTMs excel at sequence modeling and time series forecasting tasks.
• Real-time analysis and monitoring using predictive analytics and anomaly detection over streaming data. Real-time analytics will become indispensable for time-critical monitoring and decision-making applications as computational speeds increase.
• Automated time series model selection using hyperparameter tuning, Bayesian methods, genetic algorithms, and other techniques to systematically determine the optimal model specifications and parameters for a given dataset and context. This relieves analysts of much tedious trial-and-error testing.
• State-of-the-art missing data imputation, cleaning, and preprocessing techniques to overcome data quality issues: For example, advanced interpolation, Kalman filtering, and robust statistical methods can minimize distortions caused by gaps, noise, outliers, and irregular intervals in time series data.

In summary, we can expect major leaps in time series forecasting accuracy, efficiency, and applicability as modern AI and data processing innovations integrate into standard applied analytics practice. The future is bright for leveraging these technologies to extract valuable insights from time series data.

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C4w1: working with time series, c4w1: working with time series #.

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Course info.

• Prof. Anna Mikusheva

Departments

As taught in.

• Probability and Statistics
• Econometrics
• Macroeconomics

Learning Resource Types

Time series analysis, assignments and exams.

The assignments and exam from the course are not available, however some sample problems and a sample final exam have been included.

Problem Sets

The problem sets will emphasize different aspects of the course, including theory and estimation procedures we discuss in class. I strongly believe that the best way to learn the techniques is by doing. Every problem set will include an applied task that may include computer programming. I do not restrict you in your choice of computer language. I also do not require you to write all programs by yourself from scratch. You may use user-written parts of codes you find on the Internet, but I do require that you understand the program you use and properly document it with all needed citations of original sources. Collaboration with other students on problem sets is encouraged, however, the problem sets should be written independently.

Samples problems for 14.384 (PDF)

Sample final exam (PDF)

You are leaving MIT OpenCourseWare

Time Series and Forecasting: A Project-based Approach with R

Chapter 21 assignment: project proposal.

In this assignment you will develop your initial concept note into a draft of a full project proposal. Treat this assignment as a “dry run” for developing a proposal for a grant or fellowship application, or for your Ph.D. prospectus.

Your proposal should include at least the following sections and information.

Front matter: Descriptive title, your name, date, reference to “SYS 5581 Time Series & Forecasting, Spring 2021”.

Abstract: A very brief summary of the project.

21.1 Introduction

Give a narrative description of the problem you are addressing, and the methods you will use to address it. Provide context:

• What is the question you are attempting to answer?
• Why is this question important? (Who cares?)
• How will you go about attempting to answer this question?

This work addresses the question: Why do people not use probabilistic forecasts for decision-making?

21.2 The data and the data-generating process

Describe the data set you will be analyzing, and where it comes from, how it was generated and collected. Identify the source of the data. Give a narrative description of the data-generating process: this piece is critical.

Since these will be time series data: identify the frequency of the data series (e.g., hourly, monthly), and the period of record.

21.3 Exploratory data analysis

Provide a brief example of the data, showing how they are structured.

21.4 Plot the time series.

21.5 perform and report the results of other exploratory data analysis, 21.5.1 stl decomposition.

21.5.2 Fitting data to simple models

21.5.3 Work with ln(GDP)

21.6 Statistical model

21.6.1 formal model of data-generating process.

Write down an equation (or set of equations) that represent the data-generating process formally.

If applicable: describe any transformations of the data (e.g., differencing, taking logs) you need to make to get the data into a form (e.g., linear) ready for numerical analysis.

What kind of process is it? \(AR(p)\) ? White noise with drift? Something else?

Write down an equation expressing each realization of the stochastic process \(y_t\) as a function of other observed data (which could include lagged values of \(y\) ), unobserved parameters ( \(\beta\) ), and an error term ( \(\varepsilon_t\) ). Ex:

\[y = X\cdot\beta + \varepsilon\] Add a model of the error process. Ex: \(\varepsilon \sim N(0, \sigma^2 I_T)\) .

21.6.2 Discussion of the statistical model

Describe how the formal statistical model captures and aligns with the narrative of the data-generating process. Flag any statistical challenges raised by the data generating process, e.g. selection bias; survivorship bias; omitted variables bias, etc.

21.7 Plan for data analysis

Describe what information you wish to extract from the data. Do you wish to… estimate the values of the unobserved model parameters? create a tool for forecasting? estimate the exceedance probabilities for future realizations of \(y_t\) ?

Describe your plan for getting this information. OLS regression? Some other statistical technique?

If you can: describe briefly which computational tools you will use (e.g., R), and which packages you expect to draw on.

21.8 Submission requirements

Prepare your proposal using Markdown . (You may find it useful to generate your Markdown file from some other tool, e.g. R Markdown in R Studio.) Submit your proposal by pushing it to your repo within the course organization on Github. When your proposal is ready, notify the instructor by also creating a submission for this assignment on Collab. Please also upload a PDF version of your proposal to Collab as part of your submission.

21.9 Comment

Depending on your prior experience, you may find this assignment challenging. Treat this assignment as an opportunity to make progress on your own research program. Make your proposal as complete as you can. But note that this assignment is merely the First Draft. You will have more opportunity to refine your work over the next two months, in consultation with the instructor, your advisor, and your classmates.

21.10 References

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Time Series Analysis - Assignment 2

Time series analysis of antarctica land ice mass series, james angus, required packages, introduction, aims/objectives.

This report summarises the analysis of the Antarctica land ice mass series, and the process followed to identify suitable ARIMA(p,d,q) models as part of Assignment 2 for Time Series Analysis. The report covers analysis of the dataset, addressing non-stationarity, and using model specification tools to suggest possible ARIMA models.

Methodology

Creating a function for producing acf and pacf plots.

To reduce repetition of R codes, a function was created to produce ACF and PACF plots.

Retrieving the data

A summary of the Antarctica land ice mass data imported into R showed the imported dataset had 19 observations, including a minimum date of 2002 and a maximum date of 2020, the same as the original data file.

The data summary also showed that between 2002 and 2020, the mean annual change in land ice mass relative to 2001 was -1,063.77 billion metric tons. Considering the timeframe for the series was 19 years, it was shocking to note that the ‘Annual_ice_mass’ had a range of 2,531.51 billion metric tons, more than double the mean annual change and indicative of a very large change in land ice mass over the course of less than 20 years.

As the Antarctic land Ice dataset had been stored as a dataframe when it was first imported into R, it was converted to a time series object with a frequency of 1 as each observation was an annual change.

Exploring the data

The Antarctic land ice series was plotted to visually inspect the series for the 5 key time series traits; trend, seasonality, changing variance, behaviour, and change point/intervention.

Figure 1: Time series plot of the Annual Antarctica land ice mass series.

The time series plot of the Antarctica land ice mass series showed the following time series traits;

From the time series plot in Figure 1, it was clear the series had a downward, linear trend, indicating that the Antarctic land ice mass was getting smaller relative to 2001 as time progressed.

• Seasonality

There were no clear signs of seasonality in Figure 1.

• Changing variance

There were no clear signs of changing variance in Figure 1.

The time series plot in Figure 1 exhibited autoregressive (AR) behaviour, with many successive points. There were no clear fluctuations in the series, so there was no evidence of moving average (MA) behaviour.

• Change point

There was no clear evidence of a change point (intervention) in the Antarctica land ice mass series as the series appeared to exhibit the same trend and behaviour, with no sudden changes. This was not surprising considering the factors that would affect Antarctic land ice mass (e.g., climate change) do not change substantially from year to year. It would take a very large intervention (e.g., a meteor strike) to change the behaviour of the series.

The points in Figure 1 show evidence of succeeding measurements being related to one another, and Figure 2 highlights this relationship more clearly using a scatter plot of neighbouring pairs.

Figure 2: Scatter plot of each Annual Antarctica land ice mass measurement and the measurement taken in the previous year.

The scatter plot in Figure 2 shows a strong, upward trend, indicative of a strong correlation between neighbouring pairs. The plot indicates that small annual changes in Antarctic land ice mass (relative to 2001) tend to be followed by small changes, moderate changes tend to be followed by moderate changes, and large changes tend to be followed by large changes. These observations were supported by the correlation between neighbouring annual land ice mass changes which, at 0.985, indicated a very strong, positive correlation between neighbouring points.

Assessing stationarity

Figure 1 showed a strong, downward, linear trend in the Antarctica ice series, suggesting that the raw Antarctica ice series was non-stationary. As ARIMA models can only be produced using a stationary series, ACF and PACF plotswere produced to assess the series for stationarity.

Figure 3: ACF and PACF plots of the Antarctica land ice mass series.

The ACF plot in Figure 3 showed 3 significant autocorrelation lags and a decaying pattern, indicative of a non-stationary, autoregressive process. There was no evidence of seasonality in the series as there was no clear wave pattern in the ACF lags.

The first partial autocorrelation lag in the PACF plot in Figure 3 was highly significant, while all other lags were not significant, indicative of a non-stationary, autoregressive series. Several unit root tests, including an Augmented Dickey-Fuller test, were also used to assess the series for stationarity, with results shown in Table 4.

Table 4: Statistical tests for stationarity and normality in the Antarctica land ice mass series.

The null hypothesis under the Augmented Dickey-Fuller test and the Phillips-Perron (PP) test is that the series Is non-stationary. With p-values of 0.30 for the Augmented Dickey-Fuller test, and 0.75 for the PP test, there was insufficient evidence to reject the null hypothesis of both tests and therefore we could conclude that the series was not stationary. This conclusion was supported by results of a KPSS test, which produced a p-value of 0.01, indicating there was sufficient evidence to reject the null hypothesis under the KPSS test that the series was stationary. Finally, a Q-Q plot was produced to assess the raw series for normality, alongside a Shapiro-Wilk test. Although the Q-Q plot in Figure 4 showed deviations from the diagonal in each tail of the series, the Shapiro-Wilk test produced a p-value of 0.147, indicating that the measurements in the Antarctica land ice mass series were approximately normally distributed.

Figure 4: Normal Q-Q plot of the Antarctica land ice mass series.

Table 5: Result of the Shapiro-Wilk test on the raw Antarctica Ice series.

Addressing non-stationarity

Transformation.

Although there was no clear evidence of changing variance in Figure 1, a Box-Cox transformation was applied to the series to observe the impact of transformation on the series.

As the series contained negative values (declines in the land ice mass relative to 2001), a constant had to be added to the series to make all values greater than zero before a Box-Cox transformation could be applied. A vector of candidate power transformation values was also required to obtain candidate lambda values for the Box-Cox transformation due to computational limits. Through trial and error, limits of zero and 2 were identified as appropriate power transformation limits for the Box-Cox transformation.

The plot of candidate lambda values for the Box-Cox transformation of the Antarctica land ice mass series can be seen in Figure 5. Table 6 shows the values of the first and third vertical lines from Figure 5 were 0.65 and 1.22, respectively, and the chosen lambda value for the Box-Cox transformation (the middle vertical line in Figure 5) was 0.89.

Figure 5: Log likelihood plot for the lambda values in the Box-Cox transformation of the Antarctica land ice mass series.

Table 6: Results for candidate lambda values from the Box-Cox power transformation function.

With an appropriate lambda value identified, the Box-Cox transformation was applied to the Antarctica Ice series and the resultant series was plotted for visual inspection in Figure 6.

Figure 6: Time series plot of the Box-Cox transformed Annual Antarctica land ice mass series.

The time series plot in Figure 6 showed no clear change or improvement in the variation of the series, and the Q-Q plot in Figure 7 still showed deviations from the diagonal at both tails of the series. There was also little change in the result from the Shapiro-Wilk test of the Box-Cox transformed series, shown in Table 7, indicating that values in the series were still approximately normally distributed.

Figure 7: Normal Q-Q Plot for the Box-Cox transformed Antarctica Ice series.

Table 7: Results from the Shapiro-Wilk test of the Box-Cox transformed Antarctic Ice series.

Finally, the Augmented Dickey-Fuller test (Table 8) produced a p-value of 0.563, indicating that the Box-Cox transformed Antarctica Ice series was non-stationary.

Table 8: Results from the Augmented Dickey-Fuller test of the Box-Cox transformed Antarctic Ice series.

As there was no evidence to suggest the raw Antarctica Ice series exhibited changing variation, and the Box-Cox transformation did not change or improve the normality of the series, the Box-Cox transformation was deemed unnecessary, and the raw series would be used when applying differencing to address non-stationarity.

Differencing

Differencing was applied to the raw Antarctica Ice series to address the trend in the raw series, with the resulting series plotted in Figure 8.

Figure 8: Time series plot of first differenced Antarctica Ice series

The time series plot in Figure 8 appeared to exhibit a slight downward trend, indicating that the first differenced series may still be non-stationary. However, the ACF and PACF plots of the first differenced Antarctica Ice series (Figure 9) showed no significant points and no decaying pattern, suggesting the first differenced series may have been stationary.

Figure 9: ACF and PACF plots of the first differenced Antarctica land ice mass series.

To test for stationarity, an Augmented Dickey-Fuller test of the first differenced series was performed, with the test resulting in a p-value of 0.357 which, at the α=0.05 level, indicated that the first differenced Antarctica Ice series was non-stationary.

Table 9: Result from the Augmented Dickey-Fuller test of the first differenced Antarctic Ice series.

As first differencing had not produced a stationary series, second differencing was applied to the raw Antarctica Ice series, with the resulting series plotted in Figure 10.

Figure 10: Time series plot of second differenced Antarctica Ice series.

The second differenced time series in Figure 10 showed no clear trend, suggesting that second differencing may have resulted in a stationary series.

Figure 11: ACF and PACF plots of the second differenced Antarctica land ice mass series.

The ACF and PACF plots of the second differenced series (Figure 11) showed a significant PACF lag at the second lag, suggesting the second differenced series may be non-stationary. Supporting this was the Augmented Dickey-Fuller test of the second differenced series (Table 10), which produced a p-value of 0.073, indicating that, at the α=0.05 level, the second differenced series may still have been non-stationary.

However, as the result of the Augmented Dickey-Fuller test was within the “doubt range” of 0.03 to 0.1, additional unit root tests were performed to better evaluate whether second differencing had resulted in a stationary series.

Table 10: Results from the Augmented Dickey-Fuller test, Phillips-Perron test, and KPSS test of the second differenced Antarctic Ice series.

A Phillips-Perron Unit Root test and a KPSS test (Table 11) produced p-values of 0.026, and 0.1, respectively, indicating there was sufficient evidence to conclude the second differenced series was stationary.

Table 11: Results from the Phillips-Perron test, and KPSS test of the second differenced Antarctic Ice series.

As the series appeared stationary in Figure 10, and the Phillips-Perron Unit Root test and KPSS test in Table 11 indicated the second differenced series was stationary, it was concluded that second differencing of the Antarctica Ice series had resulted in a stationary series.

As second differencing the Antarctica Ice series had resulted in a stationary series, it was deemed suitable for use in model specification. All proposed ARIMA(p,d,q) models would have a value of “d” equal to 2, as second differencing had been applied.

Model specification

Acf & pacf plots.

The first step in identifying potential models for the Antarctica Ice series was to use ACF and PACF plots to propose values for the orders of autoregression (p) and moving average (q).

Figure 12: ACF and PACF plots of the second differenced Antarctica Ice series.

From the PACF plot in Figure 12, the second partial autocorrelation lag was deemed significant, and the first lag was deemed borderline significant, indicating that the value of “p” could be 1 or 2. These values were understandable given autoregressive behaviour was identified in the raw Antarctica Ice series.

There were no clearly significant autocorrelation lags in the ACF plot of the second differenced series. However, the first lag of the ACF plot was deemed borderline significant, and thus the value of “q” was proposed as either 0 or 1.

The resulting proposed models from the ACF and PACF plots were:

• { ARIMA(1,2,0), ARIMA(2,2,0), ARIMA(1,2,1), ARIMA(2,2,1) }

The second step in identifying potential models for the Antarctica Ice series involved using the extended ACF function, or EACF. Running the EACF function on the second differenced series with default parameters resulted in an error, which was overcome by specifying the maximum values for AR and MA in the EACF function. The resulting output is shown in Table 12.

Table 12: EACF plot of the second differenced Antarctic Ice series.

The most top-left point in the EACF plot was selected as (0,0), and the models proposed using the EACF plot were:

• { ARIMA(0,2,0), ARIMA(0,2,1) }

It should be noted that the ARIMA(0,2,0) model will have no coefficients. Trend modelling would be used to see if this model captures the autocorrelation in the series.

The third step in identifying potential models for the Antarctica Ice series was to use the Bayesian Information Criterion, or “BIC”, to propose values for “p” and “q”.

The BIC plot was first produced using values of 10 for ‘nma’ and ‘nar’ however, this resulted in an error. As it was known from the ACF, PAC, and EACF plots that the orders of “p” and “q” would likely be small, the values of ‘nma’ and ‘nar’ were reduced to 5, and then to 3, with the resulting BIC plots shown in Figure 13 and Figure 14, respectively.

Figure 13: BIC plot of the second differenced Antarctic Ice series, using ‘nar’ and ‘nma’ values of 5.

Figure 14: BIC plot of the second differenced Antarctic Ice series, using ‘nar’ and ‘nma’ values of 3.

The BIC plot with values of 3 for ‘nar’ and ‘nma’ (Figure 14) proposed models that were in-line with the ACF, PACF, and EACF plots, so it was selected for use in proposing models for the second differenced series.

The model proposals from the BIC model specification method were:

• { ARIMA(1,2,1), ARIMA(2,2,1), ARIMA(3,2,1), ARIMA(1,2,0), ARIMA(2,2,0) }

Final set of possible models

The final set of possible models for the Antarctica land ice mass series was:

• { ARIMA(1,2,0),
• ARIMA(2,2,0),
• ARIMA(1,2,1),
• ARIMA(2,2,1),
• ARIMA(0,2,0),
• ARIMA(0,2,1),
• ARIMA(3,2,1) }

Most of these models made sense in the context of the data exploration, which identified autoregressive behaviour in the raw Antarctica Ice series (i.e., “p” would likely be non-zero) and no clear evidence of moving average behaviour (i.e., “q” would likely be low or zero). It should be noted that the ARIMA(0,2,0) model will have no coefficients, and trend modelling would be required to see if this model captures the autocorrelation in the series.

The following models were proposed by more than one model specification method;

• ARIMA(2,2,1) }

Further testing, including diagnostic checking, would be required to identify the optimal model from the set of proposed models.

Summary & Conclusion

The objective of the analysis and model specification conducted was to understand the Antarctica land ice mass dataset, and to propose a set of possible ARIMA(p,d,q) models for the series.

Exploration of the Antarctica land ice mass dataset revealed a large range and a mean change of -1,063.77 billion metric tons, relative to the ice mass in 2001.

Analysis of the Antarctica Ice series identified that the series exhibited a clear downward trend and autoregressive behaviour (successive points), with no clear signs of seasonality, changing variance, or a change point. A high correlation between neighbouring points was also identified, which aligned with the autocorrelation behaviour observed when the series was plotted.

ACF and PACF plots of the Antarctica Ice series identified significant autocorrelation lags, and a single, very significant partial autocorrelation lag, indicative of a non-stationary series. Further tests confirmed that the raw series was not stationary, but that the measurements in the series were approximately normally distributed.

Although the series did not exhibit changing variance, a Box-Cox transformation was applied to the series to observe its impact. Visual inspection, a Shapiro-Wilk test, and a unit root test identified that the Box-Cox transformation did not change the series much, so the transformation was deemed unnecessary.

Differencing was then applied to the raw series to address the trend in the series. Visual inspection and a unit root test confirmed that first differencing did not overcome the non-stationarity, and so second differencing was applied. Although the Augmented Dickey-Fuller test of the second differenced series suggested the series was still non-stationary, visual inspection, and further unit root tests (PP and KPSS) confirmed that second differencing had made the series stationary. As such, the second differenced series was chosen for use in model specification.

ACF, PACF, EACF and BIC model specification tools were used to proposed suitable ARIMA(p,d,q) models for the Antarctica Ice series, with the final set of proposed models containing 7 ARIMA(p,d,q) models;

{ ARIMA(1,2,0), ARIMA(2,2,0), ARIMA(1,2,1), ARIMA(2,2,1), ARIMA(0,2,0), ARIMA(0,2,1), ARIMA(3,2,1) }.

Mets' Senga says he needs more time before beginning rehab stint

N EW YORK (AP) — Kodai Senga, who has yet to pitch this season for the New York Mets due to a right shoulder capsule strain, will continue working on his mechanics before beginning a rehab assignment.

“With my current mechanics, I didn’t think I’d be able to come back at 100 percent,” Senga said via an interpreter prior to Monday’s series opener against the Philadelphia Phillies. “So taking a little bit of time to look over everything, making sure everything is perfect before I get back into games, is the right move.”

Senga said his mechanics didn’t feel right during his first live batting practice session April 29. The uncertainty continued after a second batting practice session May 4.

“All my power output was not going toward the catcher,” Senga said. “I wasn’t able to deliver 100% of it toward the catcher, which is very important. When that is happening, I’m more susceptible to getting hit and also more susceptible to injuries if that continues.”

Senga, who signed a five-year deal with the Mets in December 2022 following an 11-year career in his native Japan, and Mets manager Carlos Mendoza acknowledged the cultural differences between how injured players are treated in Japan and the United States.

“In Japan, it’s more up to the player — if the player feels good, they can keep pushing forward,” Senga said. “Here the trainers have a very well-structured program.”

Mendoza said Senga felt so good after a game of catch Sunday that he subsequently threw 45 pitches off a bullpen mound — something Senga didn’t acknowledge during his meeting with reporters a few minutes earlier, when he said he’d next throw a bullpen on Wednesday.

“It’s a unique situation,” Mendoza said. “I’m trying to learn the individual myself and trying to get to know him and some of the things that he does and he likes to do.”

Senga said he didn’t know how many bullpen sessions he would need before he could begin a rehab assignment. The 31-year-old finished second in the National League Rookie of the Year balloting after going 12-7 with a 2.98 ERA and 202 strikeouts last season but reported arm fatigue shortly after reporting to spring training in February.

“At the end of the day, you don’t want to put a player at risk, especially if he’s not feeling the way he thinks he should be feeling,” Mendoza said.

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Video of Nikola Jokic’s clever time-wasting tactics goes viral

Nikola Jokic may just have himself a future as a soccer player.

Jokic and the Denver Nuggets evened up their second-round series against the Minnesota Timberwolves at two games apiece with a 115-107 victory in Sunday’s Game 4 in Minnesota. After leading by as many as 18 at one point, the Nuggets held off to a late comeback effort by the Timberwolves to hang on for the win.

After the game, a video from the “Hot Hand Theory” podcast went viral pointing out some shrewd time-wasting tactics that Jokic engaged in during the game. Following multiple made baskets by the surging Timberwolves in the fourth quarter, Jokic cleverly bled some clock by intentionally letting the ball bounce or by delaying in giving the ball back to the referee so that the five-second inbounds timer would not begin yet.

The video noted that Jokic successfully wasted nearly a minute of game clock during the fourth quarter through his tactics. Take a look below.

No player in the NBA is as effective or blatant in stalling with a lead than Nikola Jokic. He does this by delaying the refs 5-second count (fumbling the ball or passing it to the ref) and then rolling it to his teammate. Tonight he wasted almost a minute of game clock in the 4Q. pic.twitter.com/ToKwHZooC1 — Hot Hand Theory (@HotHandTheory) May 13, 2024

Of course, Jokic is far from the first or the only NBA player to use these kinds of methods to fritter away the clock. But with Minnesota cutting the lead down to six points (two possessions) late in the fourth quarter, Jokic’s ploy appears to have worked to perfection.

The strategy does come with some obvious limitations though. It only works when a team is nursing a lead, so Jokic and the Nuggets still need to take care of business otherwise for it to come into play. But this is yet another reason why the newly-minted three-time MVP Jokic is one of the most clever players in the league right now .

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The Cleveland Cavaliers will be without their leader for their most important game of the season thus far. Cleveland announced on Monday that All-Star guard Donovan Mitchell will miss Game 4 against the Boston Celtics because of a strain to…

Tom Brady’s first assignment for broadcasting debut with FOX revealed

Tom Brady’s first game assignment for his anticipated broadcasting debut with FOX has been revealed. The NFL on Monday announced a few games that will take place in Week 1 of the 2024 season. The full regular season schedule is…

Nikola Jokic may just have himself a future as a soccer player. Jokic and the Denver Nuggets evened up their second-round series against the Minnesota Timberwolves at two games apiece with a 115-107 victory in Sunday’s Game 4 in Minnesota.…