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This paper develops new mathematical techniques to identify temporal shifts among a collection of US equities partitioned into a new and more detailed set of market sectors. Although conceptually related, our three analyses reveal distinct insights about financial markets, with meaningful implications for investment managers. First, we explore a variety of methods to identify nonlinear shifts in market sector structure and describe the mathematical connection between the measure used and the captured phenomena. Second, we study network structure with respect to our new market sectors and identify meaningfully connected sector-to-sector mappings. Finally, we conduct a series of sampling experiments over different sample spaces and contrast the distribution of Sharpe ratios produced by long-only, long-short and short-only investment portfolios. In addition, we examine the sector composition of the top-performing portfolios for each of these portfolio styles. In practice, the methods proposed in this paper could be used to identify regime shifts, optimally structured portfolios, and better communities of equities., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

We introduce new mathematical methods to study the optimal portfolio size of investment portfolios over time, considering investors with varying skill levels. First, we explore the benefit of portfolio diversification on an annual basis for poor, average and strong investors defined by the 10th, 50th and 90th percentiles of risk-adjusted returns, respectively. Second, we conduct a thorough regression experiment examining quantiles of risk-adjusted returns as a function of portfolio size across investor ability, testing for trends and curvature within these functions. Finally, we study the optimal portfolio size for poor, average and strong investors in a continuously temporal manner using more than 20 years of data. We show that strong investors should hold concentrated portfolios, poor investors should hold diversified portfolios; average investors have a less obvious distribution with the optimal number varying materially over time., Comment: Minor edits since v2. Equal contribution

In this paper we contrast the dynamics of the 2022 Ukraine invasion financial crisis with notable financial crises of the 21st century - the dot-com bubble, global financial crisis and COVID-19. We study the similarity in market dynamics and associated implications for equity investors between various financial market crises and we introduce new mathematical techniques to do so. First, we study the strength of collective dynamics during different market crises, and compare suitable portfolio diversification strategies with respect to the unique number of sectors and stocks for optimal systematic risk reduction. Next, we introduce a new linear operator method to quantify distributional distance between equity returns during various crises. Our method allows us to fairly compare underlying stock and sector performance during different time periods, normalising for those collective dynamics driven by the overall market. Finally, we introduce a new combinatorial portfolio optimisation framework driven by random sampling to investigate whether particular equities and equity sectors are more effective in maximising investor risk-adjusted returns during market crises., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

Since its conception, the cryptocurrency market has been frequently described as an immature market, characterized by significant swings in volatility and occasionally described as lacking rhyme or reason. There has been great speculation as to what role it plays in a diversified portfolio. For instance, is cryptocurrency exposure an inflationary hedge or a speculative investment that follows broad market sentiment with amplified beta? We have recently explored similar questions with a clear focus on the equity market. There, our research revealed several noteworthy dynamics such as: an increase in the market's collective strength and uniformity during crises, greater diversification benefits across equity sectors (rather than within them), and the existence of a "best value" portfolio of equities. In essence, we can now contrast any potential signatures of maturity we identify in the cryptocurrency market and contrast these with the substantially larger, older and better established equity market. This paper aims to investigate whether the cryptocurrency market has recently exhibited similar mathematical properties as the equity market. Instead of relying on traditional portfolio theory, which is grounded in the financial dynamics of equity securities, we adjust our experimental focus to capture the presumed behavioral purchasing patterns of retail cryptocurrency investors. Our focus is on collective dynamics and portfolio diversification in the cryptocurrency market, and examining whether previously established results in the equity market hold in the cryptocurrency market, and to what extent. Results reveal nuanced signatures of maturity related to the equity market, including the fact that correlations collectively spike around exchange collapses, and identify an ideal portfolio size and spread across different groups of cryptocurrencies., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

On February 28, 2021, a fireball dropped $\sim0.6$ kg of recovered CM2 carbonaceous chondrite meteorites in South-West England near the town of Winchcombe. We reconstruct the fireball's atmospheric trajectory, light curve, fragmentation behaviour, and pre-atmospheric orbit from optical records contributed by five networks. The progenitor meteoroid was three orders of magnitude less massive ($\sim13$ kg) than any previously observed carbonaceous fall. The Winchcombe meteorite survived entry because it was exposed to a very low peak atmospheric dynamic pressure ($\sim0.6$ MPa) due to a fortuitous combination of entry parameters, notably low velocity (13.9 km/s). A near-catastrophic fragmentation at $\sim0.07$ MPa points to the body's fragility. Low entry speeds which cause low peak dynamic pressures are likely necessary conditions for a small carbonaceous meteoroid to survive atmospheric entry, strongly constraining the radiant direction to the general antapex direction. Orbital integrations show that the meteoroid was injected into the near-Earth region $\sim0.08$ Myr ago and it never had a perihelion distance smaller than $\sim0.7$ AU, while other CM2 meteorites with known orbits approached the Sun closer ($\sim0.5$ AU) and were heated to at least 100 K higher temperatures., Comment: Accepted for publication in MAPS

This paper uses established and recently introduced methods from the applied mathematics and statistics literature to study trends in the end-use sector and capacity of low-carbon hydrogen projects in recent and upcoming decades. First, we examine distributions in plants over time for various end-use sectors and classify them according to metric discrepancy, observing clear similarity across all industry sectors. Next, we compare the distribution of usage sectors among different continents and examine the changes in sector distribution over time. Finally, we judiciously apply several regression models to analyse the association between various predictors and the capacity of global hydrogen projects. Across our experiments, we see a welcome exponential growth in the capacity of zero-carbon hydrogen plants and significant growth of new and planned hydrogen plants in the 2020's across every sector., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

This paper introduces new methods for studying the prevalence of terrorism around the world and over time. Our analysis treats spatial prevalence of terrorism, the changing profile of groups carrying out the acts of terrorism, and trends in how many attacks take place over time. First, we use a time-evolving cluster analysis to show that the geographic distribution of regions of high terrorist activity remains relatively consistent over time. Secondly, we use new metrics, inspired by geometry and probability, to track changes in the distributions of which groups are performing the terrorism. We identify times at which this distribution changes significantly and countries where the time-varying breakdown is most and least homogeneous. We observe startling geographic patterns, with the greatest heterogeneity from Africa. Finally, we use a new implementation of distances between distributions to group countries according to their incidence profiles over time. This analysis can aid in highlighting structural similarities in outbreaks of extreme behavior and the most and least significant public policies in minimizing a country's terrorism., Comment: Accepted manuscript. New analysis and minor text edits since v1

This paper applies new and recently introduced approaches to study trends in gun violence in the United States. We use techniques in both the time and frequency domain to provide a more complete understanding of gun violence dynamics. We analyze gun violence incidents on a state-by-state basis as recorded by the Gun Violence Archive. We have numerous specific phenomena of focus, including periodicity of incidents, locations in time where behavioral changes occur, and shifts in gun violence patterns since April 2020. First, we implement a recently introduced method of spectral density estimation for nonstationary time series to investigate periodicity on a state-by-state basis, including revealing where periodic behaviors change with time. We can also classify different patterns of behavioral changes among the states. We then aim to understand the most significant shifts in gun violence since numerous key events in 2020, including the COVID-19 pandemic, lockdowns, and periods of civil unrest. Our dual-domain analysis provides a more thorough understanding and challenges numerous widely held conceptions regarding the prevalence of gun violence incidents., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

This paper uses new and existing methods to study collective trends across countries throughout the pandemic, with a focus on the multivariate time series of reproduction numbers and vaccine proliferation. We begin with a time-varying analysis of the collective nature of infectivity, where we evaluate the eigenspectrum and collective magnitude of reproduction number time series on a country-by-country basis. Next, we study the topology of this eigenspectrum, measuring the deviation between all points in time, and introduce a graph-theoretic methodology to reveal a clear partition in global infectivity dynamics. Then, we compare countries' vaccine rollouts with economic indicators such as their GDP and HDI in a collective fashion. We investigate time-varying consistency and determine points in time where there is the greatest discrepancy between these indicators as a whole. Our two primary findings are a considerable increase in collective infectivity in the latter half of the period, and a concave-up (``down then up'') pattern in the collective consistency between vaccine coverage and economic/development indicators across countries., Comment: Accepted manuscript. Moderate edits since v1. Equal contribution

Motivated by the current fears of a potentially stagflationary global economic environment, this paper uses new and recently introduced mathematical techniques to study multivariate time series pertaining to country inflation (CPI), economic growth (GDP) and equity index behaviours. We begin by assessing the temporal evolution among various economic phenomena, and complement this analysis with `economic driver analysis,' where we decouple country economic trajectories and determine what is most important in their association. Next, we study the temporal self-similarity of global inflation, growth and equity index returns to identify the most anomalous historic periods, and windows in the past that are most similar to current market dynamics. We then introduce a new algorithm to construct economic state classifications and compute an economic state integral, where countries are determined to belong in one of four candidate states based on their inflation and growth behaviours. Finally, we implement a decade-by-decade portfolio optimisation to determine which equity indices and portfolio assets have been most beneficial in maximising portfolio risk-adjusted returns in various market conditions. This could be of great interest to those looking for asset allocation guidance in the current period of high economic uncertainty., Comment: Accepted manuscript. Minor edits since v3

We study how to assess the potential benefit of diversifying an equity portfolio by investing within and across equity sectors. We analyse 20 years of US stock price data, which includes the global financial crisis (GFC) and the COVID-19 market crash, as well as periods of financial stability, to determine the `all weather' nature of equity portfolios. We establish that one may use the leading eigenvalue of the cross-correlation matrix of log returns as well as graph-theoretic diagnostics such as modularity to quantify the collective behaviour of the market or a subset of it. We confirm that financial crises are characterised by a high degree of collective behaviour of equities, whereas periods of financial stability exhibit less collective behaviour. We argue that during times of increased collective behaviour, risk reduction via sector-based portfolio diversification is ineffective. Using the degree of collectivity as a proxy for the benefit of diversification, we perform an extensive sampling of equity portfolios to confirm the old financial adage that 30-40 stocks provide sufficient diversification. Using hierarchical clustering, we discover a `best value' equity portfolio for diversification consisting of 36 equities sampled uniformly from 9 sectors. We further show that it is typically more beneficial to diversify across sectors rather than within. Our findings have implications for cost-conscious retail investors seeking broad diversification across equity markets., Comment: Accepted manuscript. Minor edits and typos fixed since v1. Equal contribution

We introduce new frameworks to study spatio-temporal patterns in carbon dioxide emissions, demographic trends and economic patterns across 50 countries over the past 50 years. Our analysis is broken up into four sections. First, we introduce a new method to classify countries into one of three characteristic emissions classes based on a one, two or three-segment piecewise linear model. We reveal that most countries are best represented by a piecewise linear model with one change point. Next, we perform a decade-by-decade study of carbon dioxide trajectories. There, we demonstrate notable changes in cluster structures in each decade. We then study the spatial propagation of emissions over time, highlighting a peak in spatial dispersion in 2000, beyond which there has been a gradual decline in spatial emissions variance over space. Finally, we use carbon dioxide, GDP, and population data and apply dimensionality reduction and clustering to group countries based on similarity in their real and carbon economies., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

This paper uses new and recently established methodologies to study the evolutionary dynamics of the cryptocurrency market, and compares the findings with that of the equity market. We begin by applying random matrix theory and principal components analysis (PCA) to correlation matrices of both collections, highlighting clear differences in the eigenspectra exhibited. We then explore the heterogeneity of both asset classes, studying the time-varying dynamics of underlying sector behaviours, and determine the collective similarity within each collection. We then turn to a study of structural break dynamics and evolutionary power spectra, where we quantify the collective affinity in structural breaks and evolutionary behaviours of underlying sector time series. Finally, we implement two algorithms simulating `portfolio choice' dynamics to compare the effectiveness of stock selection and sector allocation in cryptocurrency portfolios. There, we highlight the importance of both endeavours and comment on noteworthy implications for cryptocurrency portfolio management., Comment: 45 pages

We use new and established methodologies in multivariate time series analysis to study the dynamics of 414 Australian hydrological stations' streamflow. First, we analyze our collection of time series in the temporal domain, and compare the similarity in hydrological stations' candidate trajectories. Then, we introduce a Whittle Likelihood-based optimization framework to study the collective similarity in periodic phenomena among our collection of stations. Having identified noteworthy similarity in the temporal and spectral domains, we introduce an algorithmic procedure to estimate a governing hydrological streamflow process across Australia. To determine the stability of such behaviours over time, we then study the evolution of the governing dynamics and underlying time series with time-varying applications of principal components analysis (PCA) and spectral analysis., Comment: 29 pages

This paper uses new and recently introduced mathematical techniques to undertake a data-driven study on the systemic nature of global inflation. We start by investigating country CPI inflation over the past 70 years. There, we highlight the systemic nature of global inflation with a judicious application of eigenvalue analysis and determine which countries exhibit most "centrality" with an inner-product based optimization method. We then turn to inflationary impacts on financial market securities, where we explore country equity indices' equity robustness and the varied performance of equity sectors during periods of significant inflationary pressure. Finally, we implement a time-varying portfolio optimization to determine which asset classes were most beneficial in increasing portfolio Sharpe ratio when an investor must hold a core (and constant) allocation to equities., Comment: 27 pages

This paper introduces new methods to track the offset between two multivariate time series on a continuous basis. We then apply this framework to COVID-19 counts on a state-by-state basis in the United States to determine the progression from cases to deaths as a function of time. Across multiple approaches, we reveal an "up-down-up" pattern in the estimated offset between reported cases and deaths as the pandemic progresses. This analysis could be used to predict imminent increased load on a healthcare system and aid the allocation of additional resources in advance., Comment: Final version. Minor edits since v1. Equal contribution

This paper uses established and recently introduced methods from the applied mathematics and statistics literature to study trends in the propagation and capacity of low-carbon hydrogen projects over the past two decades. First, we judiciously apply a regression model to estimate the association between various predictors and the capacity of global hydrogen projects. Next, we turn to the geographic propagation of low-carbon hydrogen projects, where we apply a recently introduced method to explore the geographic variance of hydrogen projects over time. Then, we demonstrate that most geographic regions display linear growth in cumulative plants and apply distance correlation to determine the nonlinear dependence between the two most prolific regions - North America and Europe. Finally, we study the time-varying regional consistency between the contribution of green vs fossil fuel plants to the total number and capacity of hydrogen plants., Comment: Accepted manuscript. Minor edits since v1. Equal contribution

This paper applies existing and new approaches to study trends in the performance of elite athletes over time. We study both track and field scores of men and women athletes on a yearly basis from 2001 to 2019, revealing several trends and findings. First, we perform a detailed regression study to reveal the existence of an "Olympic effect", where average performance improves during Olympic years. Next, we study the rate of change in athlete performance and fail to reject the notion that athlete scores are leveling off, at least among the top 100 annual scores. Third, we examine the relationship in performance trends among men and women's categories of the same event, revealing striking similarity, together with some anomalous events. Finally, we analyze the geographic composition of the world's top athletes, attempting to understand how the diversity by country and continent varies over time across events. We challenge a widely held conception of athletics, that certain events are more geographically dominated than others. Our methods and findings could be applied more generally to identify evolutionary dynamics in group performance and highlight spatio-temporal trends in group composition., Comment: Numerous new experiments since v1. Equal contribution from first two authors

This paper compares and contrasts the spread and impact of COVID-19 in the three countries most heavily impacted by the pandemic: the United States (US), India and Brazil. All three of these countries have a federal structure, in which the individual states have largely determined the response to the pandemic. Thus, we perform an extensive analysis of the individual states of these three countries to determine patterns of similarity within each. First, we analyse structural similarity and anomalies in the trajectories of cases and deaths as multivariate time series. Next, we study the lengths of the different waves of the virus outbreaks across the three countries and their states. Finally, we investigate suitable time offsets between cases and deaths as a function of the distinct outbreak waves. In all these analyses, we consistently reveal more characteristically distinct behaviour between US and Indian states, while Brazilian states exhibit less structure in their wave behaviour and changing progression between cases and deaths., Comment: Final version. Numerous edits since v2. Equal contribution from first two authors

This paper introduces a novel approach to spatio-temporal data analysis using metric geometry to study the propagation of COVID-19 across the United States. Using a geodesic Wasserstein metric, we analyse discrepancies between the density functions of new case counts on any given day, incorporating the geographic spread of cases. First, we apply this to identify the periods during which the changes in the geographic distribution of COVID-19 were most profound. The greatest shift occurred between May and June of 2020, when COVID-19 shifted from mostly dominating the Northeastern states to a wider distribution across the country. We support our findings with a new measure of the extent of geodesic variance of a distribution, demonstrating that the geographic imprint of COVID-19 was most concentrated in May 2020. Next, we investigate whether the epidemic exhibited meaningful patterns of spatial reversion, where similar geographic distributions return later. We identify broad similarity between the spread of COVID-19 across the US between the second and third waves, and to a lesser extent, the reemergence of the first wave's Northeastern dominance closer to the present day. This methodology could provide new insights for analysts to monitor the dynamical spread of epidemics and enable regional policymakers to protect their localities. More broadly, the framework we introduce could be applied to a variety of problems evolving over space and time., Comment: Equal contribution from first two authors

This paper introduces new methods to study behaviours among the 52 largest cryptocurrencies between 01-01-2019 and 30-06-2021. First, we explore evolutionary correlation behaviours and apply a recently proposed turning point algorithm to identify regimes in market correlation. Next, we inspect the relationship between collective dynamics and the cryptocurrency market size - revealing an inverse relationship between the size of the market and the strength of collective dynamics. We then explore the time-varying consistency of the relationships between cryptocurrencies' size and their returns and volatility. There, we demonstrate that there is greater consistency between size and volatility than size and returns. Finally, we study the spread of volatility behaviours across the market changing with time by examining the structure of Wasserstein distances between probability density functions of rolling volatility. We demonstrate a new phenomenon of increased uniformity in volatility during market crashes, which we term \emph{volatility dispersion}., Comment: Final version. Moderate edits since v1. Equal contribution

This paper introduces a new framework to quantify distance between finite sets with uncertainty present, where probability distributions determine the locations of individual elements. Combining this with a Bayesian change point detection algorithm, we produce a new measure of similarity between time series with respect to their structural breaks. First, we demonstrate the algorithm's effectiveness on a collection of piecewise autoregressive processes. Next, we apply this to financial data to study the erratic behavior profiles of 19 countries and 11 sectors over the past 20 years. Our measure provides quantitative evidence that there is greater collective similarity among sectors' erratic behavior profiles than those of countries, which we observe upon individual inspection of these time series. Our measure could be used as a new framework or complementary tool for investors seeking to make asset allocation decisions for financial portfolios., Comment: Accepted manuscript. Substantial edits since v1. Equal contribution

This paper investigates the relationship between the spread of the COVID-19 pandemic, the state of community activity, and the financial index performance across 20 countries. First, we analyze which countries behaved similarly in 2020 with respect to one of three multivariate time series: daily COVID-19 cases, Apple mobility data and national equity index price. Next, we study the trajectories of all three of these attributes in conjunction to determine which exhibited greater similarity. Finally, we investigate whether country financial indices or mobility data responded quicker to surges in COVID-19 cases. Our results indicate that mobility data and national financial indices exhibited the most similarity in their trajectories, with financial indices responding quicker. This suggests that financial market participants may have interpreted and responded to COVID-19 data more efficiently than governments. Further, results imply that efforts to study community mobility data as a leading indicator for financial market performance during the pandemic were misguided., Comment: Substantial edits and new experiments since v1. Equal contribution

This paper introduces new methods to study the changing dynamics of COVID-19 cases and deaths among the 50 worst-affected countries throughout 2020. First, we analyse the trajectories and turning points of rolling mortality rates to understand at which times the disease was most lethal. We demonstrate five characteristic classes of mortality rate trajectories and determine structural similarity in mortality trends over time. Next, we introduce a class of \emph{virulence matrices} to study the evolution of COVID-19 cases and deaths on a global scale. Finally, we introduce \emph{three-way inconsistency analysis} to determine anomalous countries with respect to three attributes: countries' COVID-19 cases, deaths and human development indices. We demonstrate the most anomalous countries across these three measures are Pakistan, the United States and the United Arab Emirates., Comment: Equal contribution. Substantial edits and more discussion relative to v1

This paper uses new and recently introduced methodologies to study the similarity in the dynamics and behaviours of cryptocurrencies and equities surrounding the COVID-19 pandemic. We study two collections; 45 cryptocurrencies and 72 equities, both independently and in conjunction. First, we examine the evolution of cryptocurrency and equity market dynamics, with a particular focus on their change during the COVID-19 pandemic. We demonstrate markedly more similar dynamics during times of crisis. Next, we apply recently introduced methods to contrast trajectories, erratic behaviours, and extreme values among the two multivariate time series. Finally, we introduce a new framework for determining the persistence of market anomalies over time. Surprisingly, we find that although cryptocurrencies exhibit stronger collective dynamics and correlation in all market conditions, equities behave more similarly in their trajectories, extremes, and show greater persistence in anomalies over time., Comment: As published in Physica A

This paper introduces new methods to analyze the changing progression of COVID-19 cases to deaths in different waves of the pandemic. First, an algorithmic approach partitions each country or state's COVID-19 time series into a first wave and subsequent period. Next, offsets between case and death time series are learned for each country via a normalized inner product. Combining these with additional calculations, we can determine which countries have most substantially reduced the mortality rate of COVID-19. Finally, our paper identifies similarities in the trajectories of cases and deaths for European countries and U.S. states. Our analysis refines the popular conception that the mortality rate has greatly decreased throughout Europe during its second wave of COVID-19; instead, we demonstrate substantial heterogeneity throughout Europe and the U.S. The Netherlands exhibited the largest reduction of mortality, a factor of 16, followed by Denmark, France, Belgium, and other Western European countries, greater than both Eastern European countries and U.S. states. Some structural similarity is observed between Europe and the United States, in which Northeastern states have been the most successful in the country. Such analysis may help European countries learn from each other's experiences and differing successes to develop the best policies to combat COVID-19 as a collective unit., Comment: Accepted manuscript. New appendix relative to v1

This paper analyzes the impact of COVID-19 on the populations and equity markets of 92 countries. We compare country-by-country equity market dynamics to cumulative COVID-19 case and death counts and new case trajectories. First, we examine the multivariate time series of cumulative cases and deaths, particularly regarding their changing structure over time. We reveal similarities between the case and death time series, and key dates that the structure of the time series changed. Next, we classify new case time series, demonstrate five characteristic classes of trajectories, and quantify discrepancy between them with respect to the behavior of waves of the disease. Finally, we show there is no relationship between countries' equity market performance and their success in managing COVID-19. Each country's equity index has been unresponsive to the domestic or global state of the pandemic. Instead, these indices have been highly uniform, with most movement in March., Comment: Accepted manuscript. Minor revisions compared to v1. Equal contribution

This paper introduces a mathematical framework for determining second surge behavior of COVID-19 cases in the United States. Within this framework, a flexible algorithmic approach selects a set of turning points for each state, computes distances between them, and determines whether each state is in (or over) a first or second surge. Then, appropriate distances between normalized time series are used to further analyze the relationships between case trajectories on a month-by-month basis. Our algorithm shows that 31 states are experiencing second surges, while 4 of the 10 largest states are still in their first surge, with case counts that have never decreased. This analysis can aid in highlighting the most and least successful state responses to COVID-19., Comment: Accepted manuscript. Extra appendices and resized figures compared to v1. Equal contribution

This paper proposes a cluster-based method to analyze the evolution of multivariate time series and applies this to the COVID-19 pandemic. On each day, we partition countries into clusters according to both their case and death counts. The total number of clusters and individual countries' cluster memberships are algorithmically determined. We study the change in both quantities over time, demonstrating a close similarity in the evolution of cases and deaths. The changing number of clusters of the case counts precedes that of the death counts by 32 days. On the other hand, there is an optimal offset of 16 days with respect to the greatest consistency between cluster groupings, determined by a new method of comparing affinity matrices. With this offset in mind, we identify anomalous countries in the progression from COVID-19 cases to deaths. This analysis can aid in highlighting the most and least significant public policies in minimizing a country's COVID-19 mortality rate., Comment: Accepted manuscript. Major expression edits and minor new inferences relative to v2. Equal contribution

We develop a new method to find the number of volatility regimes in a nonstationary financial time series by applying unsupervised learning to its volatility structure. We use change point detection to partition a time series into locally stationary segments and then compute a distance matrix between segment distributions. The segments are clustered into a learned number of discrete volatility regimes via an optimization routine. Using this framework, we determine a volatility clustering structure for financial indices, large-cap equities, exchange-traded funds and currency pairs. Our method overcomes the rigid assumptions necessary to implement many parametric regime-switching models, while effectively distilling a time series into several characteristic behaviours. Our results provide significant simplification of these time series and a strong descriptive analysis of prior behaviours of volatility. Finally, we create and validate a dynamic trading strategy that learns the optimal match between the current distribution of a time series and its past regimes, thereby making online risk-avoidance decisions in the present., Comment: Accepted manuscript

This article improves on existing methods to estimate the spectral density of stationary and nonstationary time series assuming a Gaussian process prior. By optimising an appropriate eigendecomposition using a smoothing spline covariance structure, our method more appropriately models data with both simple and complex periodic structure. We further justify the utility of this optimal eigendecomposition by investigating the performance of alternative covariance functions other than smoothing splines. We show that the optimal eigendecomposition provides a material improvement, while the other covariance functions under examination do not, all performing comparatively well as the smoothing spline. During our computational investigation, we introduce new validation metrics for the spectral density estimate, inspired from the physical sciences. We validate our models in an extensive simulation study and demonstrate superior performance with real data., Comment: New experiments and contributions since v2. Equal contribution

This paper introduces a new framework of algebraic equivalence relations between time series and new distance metrics between them, then applies these to investigate the Australian ``Black Summer'' bushfire season of 2019-2020. First, we introduce a general framework for defining equivalence between time series, heuristically intended to be equivalent if they differ only up to noise. Our first specific implementation is based on using change point algorithms and comparing statistical quantities such as mean or variance in stationary segments. We thus derive the existence of such equivalence relations on the space of time series, such that the quotient spaces can be equipped with a metrizable topology. Next, we illustrate specifically how to define and compute such distances among a collection of time series and perform clustering and additional analysis thereon. Then, we apply these insights to analyze air quality data across New South Wales, Australia, during the 2019-2020 bushfires. There, we investigate structural similarity with respect to this data and identify locations that were impacted anonymously by the fires relative to their location. This may have implications regarding the appropriate management of resources to avoid gaps in the defense against future fires., Comment: Minor edits since v2. Equal contribution

This paper proposes a new method for financial portfolio optimization based on reducing simultaneous asset shocks across a collection of assets. This may be understood as an alternative approach to risk reduction in a portfolio based on a new mathematical quantity. First, we apply recently introduced semi-metrics between finite sets to determine the distance between time series' structural breaks. Then, we build on the classical portfolio optimization theory of Markowitz and use this distance between asset structural breaks for our penalty function, rather than portfolio variance. Our experiments are promising: on synthetic data, we show that our proposed method does indeed diversify among time series with highly similar structural breaks and enjoys advantages over existing metrics between sets. On real data, experiments illustrate that our proposed optimization method performs well relative to nine other commonly used options, producing the second-highest returns, the lowest volatility, and second-lowest drawdown. The main implication for this method in portfolio management is reducing simultaneous asset shocks and potentially sharp associated drawdowns during periods of highly similar structural breaks, such as a market crisis. Our method adds to a considerable literature of portfolio optimization techniques in econometrics and could complement these via portfolio averaging., Comment: Accepted manuscript. Substantial additions since v2. Equal contribution from first two authors

This paper introduces new methods for analysing the extreme and erratic behaviour of time series to evaluate the impact of COVID-19 on cryptocurrency market dynamics. Across 51 cryptocurrencies, we examine extreme behaviour through a study of distribution extremities, and erratic behaviour through structural breaks. First, we analyse the structure of the market as a whole and observe a reduction in self-similarity as a result of COVID-19, particularly with respect to structural breaks in variance. Second, we compare and contrast these two behaviours, and identify individual anomalous cryptocurrencies. Tether (USDT) and TrueUSD (TUSD) are consistent outliers with respect to their returns, while Holo (HOT), NEXO (NEXO), Maker (MKR) and NEM (XEM) are frequently observed as anomalous with respect to both behaviours and time. Even among a market known as consistently volatile, this identifies individual cryptocurrencies that behave most irregularly in their extreme and erratic behaviour and shows these were more affected during the COVID-19 market crisis., Comment: Accepted manuscript. Numerous minor edits compared to v3. Equal contribution from first two authors

This paper proposes a new method for determining similarity and anomalies between time series, most practically effective in large collections of (likely related) time series, by measuring distances between structural breaks within such a collection. We introduce a class of \emph{semi-metric} distance measures, which we term \emph{MJ distances}. These semi-metrics provide an advantage over existing options such as the Hausdorff and Wasserstein metrics. We prove they have desirable properties, including better sensitivity to outliers, while experiments on simulated data demonstrate that they uncover similarity within collections of time series more effectively. Semi-metrics carry a potential disadvantage: without the triangle inequality, they may not satisfy a "transitivity property of closeness." We analyse this failure with proof and introduce an computational method to investigate, in which we demonstrate that our semi-metrics violate transitivity infrequently and mildly. Finally, we apply our methods to cryptocurrency and measles data, introducing a judicious application of eigenvalue analysis., Comment: Accepted manuscript. Minor edits since v2. Equal contribution from first two authors

Discrimination between non-stationarity and long-range dependency is a difficult and long-standing issue in modelling financial time series. This paper uses an adaptive spectral technique which jointly models the non-stationarity and dependency of financial time series in a non-parametric fashion assuming that the time series consists of a finite, but unknown number, of locally stationary processes, the locations of which are also unknown. The model allows a non-parametric estimate of the dependency structure by modelling the auto-covariance function in the spectral domain. All our estimates are made within a Bayesian framework where we use aReversible Jump Markov Chain Monte Carlo algorithm for inference. We study the frequentist properties of our estimates via a simulation study, and present a novel way of generating time series data from a nonparametric spectrum. Results indicate that our techniques perform well across a range of data generating processes. We apply our method to a number of real examples and our results indicate that several financial time series exhibit both long-range dependency and non-stationarity., Comment: 8 pages