# Enhancing financial crisis prediction: Integrating change point detection for exogenous event identification

Malvina Bozhidarova, Frank Ball, Yves van Gennip, Reuben D. O'Dea, Gilles Stupfler

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Malvina Bozhidarova, Frank Ball, Yves van Gennip, Reuben D. O'Dea, Gilles Stupfler. Enhancing financial crisis prediction: Integrating change point detection for exogenous event identification. 2025. [hal-05036063](https://hal.science/hal-05036063v1)

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Distributed under a Creative Commons Attribution 4.0 International License
Enhancing financial crisis prediction: Integrating
change point detection for exogenous event
identification

Malvina Bozhidarova¹,²*, Frank Ball², Yves van Gennip³,
Reuben O' Dea², Gilles Stupfler⁴

¹Russell Group Limited, Nottingham, NG1 1HS, UK.

²School of Mathematical Sciences, University of Nottingham,
Nottingham, NG7 2RD, UK.

³Delft Institute of Applied Mathematics (DIAM), Technische
Universiteit Delft, Delft, The Netherlands.

⁴Univ Angers, CNRS, LAREMA, SFR MATHSTIC, Angers, F-49000,
France.

*Corresponding author(s). E-mail(s): mbozhidarova@russell.co.uk;

Abstract

This paper explores the integration of change point detection (CPD) techniques to improve the adaptability of financial contagion models to major market events. Exogenous shocks, such as geopolitical tensions and natural disasters, can lead to substantial changes in stock prices and market dynamics. By implementing a real-time CPD algorithm, we enable our model to respond effectively to these disruptions, resulting in more robust and accurate predictions. We analyze stock price, geographical location, and economic sector data for a dataset of 398 companies to construct multiplex networks with four layers. On these networks, we implement a Susceptible-Infected-Recovered (SIR) transmission model to simulate the spread of financial shocks among companies, accounting for their interconnectedness. Using stock price data from the 2008 and 2020 financial crises, we evaluate the model's ability to predict the propagation of financial shocks through the network, where shocks are identified based on stock price volatility. We continuously monitor the data for anomalies and when a change point is identified, the model discards the older data before the change

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point and focuses on the more recent data. We demonstrate the effectiveness in incorporating change points for improving the model's predictive accuracy.

**Keywords:** Epidemic modelling, Financial crisis, Multiplex network, SIR model, Change point detection

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# 1 Introduction

Detecting significant changes in financial time series data is a critical aspect of quantitative analysis, especially in understanding the dynamics of market behaviour and mitigating financial risks. Financial markets are highly susceptible to various exogenous and endogenous events that can cause abrupt shifts in asset prices, volatility, trading patterns and investor decisions [1]. Accurately identifying these changes is critical for investors, financial institutions and policymakers, as it allows them to adjust their approaches in response to changing market circumstances. For instance, in the 2008 financial crisis, governments worldwide implemented various decisions and interventions to stabilize financial markets and mitigate impact of the crisis. In the United States, central banks, including the Federal Reserve, implemented aggressive interest rate cuts to stimulate economic activity and ease borrowing costs [2]. During the 2020 financial crisis, government policy regarding lockdowns, travel restrictions and quarantine measures significantly impacted various industries, leading to market disruptions, supply chain interruptions and shifts in consumer behaviour. Many sectors, such as travel, hospitality, entertainment and retail experienced substantial declines in revenue during the crisis, impacting stock prices and overall market performance [3]. In response to the market tension, governments globally introduced massive fiscal stimulus packages, providing financial support to individuals, businesses and industries severely affected by lockdowns and economic disruptions [4].

Other events, such as wars and geopolitical tensions, natural disasters and protests can also influence the financial market significantly. For example, the terrorist attacks
on September 11, 2001, led to a temporary closure of financial markets in the United States [5]. When the markets reopened, there was a significant drop in stock prices. The attacks led to increased volatility in the global financial market, particularly affecting airline-, insurance- and tourism-related stocks. Following the earthquake and tsunami in Japan in 2011, the Fukushima nuclear disaster led to widespread disruptions in Japan's economy [6]. Japanese stocks and the yen experienced severe volatility as concerns about nuclear safety, economic damage and supply chain disruptions appeared. Industries such as technology, automotive and manufacturing were particularly affected by disruptions in the supply chain [7]. Brexit, the withdrawal of the United Kingdom (UK) from the European Union (EU), also had a substantial impact on the financial markets [8]. The uncertainty surrounding Brexit negotiations led to significant fluctuations in the value of the British pound (GBP) and UK-focused stocks, particularly those reliant on EU trade or with substantial exposure to the domestic economy, were especially sensitive to Brexit developments [9].

In this paper we enhance the model introduced in [10] by incorporating change point detection (CPD) techniques in order to enable the model to react to such exogenous shocks. In [10] we introduced an innovative framework for modelling financial contagion, based on an SIR (Susceptible – Infected – Recovered) epidemic model applied to a multiplex network of interconnected companies constructed from financial data. Our model employs a stochastic epidemic transmission mechanism, allowing financial crises to propagate both locally (to neighboring nodes within the network) and globally (to any company in the system), thereby simulating how a financial shock spreads from the original infected companies to the others. The approach introduced in the current paper enhances this model to enable the dynamic integration of significant events in real-time, leading to a more robust and adaptive predictions.

The remainder of the paper is organized as follows. In Section 2 we describe the dataset and present the two components of the modelling framework: the procedure
to construct the multiplex network and the transmission mechanism. The section summarizes the methodology, introduced in detail in [10], while ensuring clarity for readers unfamiliar with the prior study. In Section 3 we build on previous work by incorporating change points into the model. We apply the model to the 2008 and the 2020 financial crises. The section begins by defining the concept of infection in a financial context and explores how the model can leverage historical data to predict future infections during each crisis. In particular, in Section 3.2 we introduce the ChangeFinder algorithm [11] and demonstrate how it can be incorporated into the model to detect significant changes and adapt to them as soon as they happen. We demonstrate the effectiveness of incorporating CPD techniques into the model by notably improving forecasts of the number of affected companies. Section 4 provides a conclusion, discusses the implications of our findings, outlines the limitations of our approach, and proposes potential areas for future research. Details of our estimation approach and further finite-sample results are provided in the Appendix.

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## 2 Modelling framework

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Our primary objective is to model the dynamics of financial contagion within a population of companies. In particular, we aim to predict the future impact of financial crises on individual companies, financial sectors and continents, leveraging historical data from past crisis days. Our modelling framework consists of two main parts. First, we construct a multiplex financial network with nodes representing individual companies and edges within each layer representing different types of connections between companies. The multiplex network approach allows us to account for the various ways in which financial contagion may propagate. Next, we apply an SIR epidemic model to each network layer. The model's key parameters are the transmission probabilities, which represent the daily likelihood of an infected company spreading the infection

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ulation of companies. In particular, we aim to predict the future impact of financial crises on individual companies, financial sectors and continents, leveraging historical data from past crisis days. Our modelling framework consists of two main parts. First, we construct a multiplex financial network with nodes representing individual companies and edges within each layer representing different types of connections between companies. The multiplex network approach allows us to account for the various ways in which financial contagion may propagate. Next, we apply an SIR epidemic model to each network layer. The model's key parameters are the transmission probabilities, which represent the daily likelihood of an infected company spreading the infection

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crises on individual companies, financial sectors and continents, leveraging historical data from past crisis days. Our modelling framework consists of two main parts. First, we construct a multiplex financial network with nodes representing individual companies and edges within each layer representing different types of connections between companies. The multiplex network approach allows us to account for the various ways in which financial contagion may propagate. Next, we apply an SIR epidemic model to each network layer. The model's key parameters are the transmission probabilities, which represent the daily likelihood of an infected company spreading the infection

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data from past crisis days. Our modelling framework consists of two main parts. First, we construct a multiplex financial network with nodes representing individual companies and edges within each layer representing different types of connections between companies. The multiplex network approach allows us to account for the various ways in which financial contagion may propagate. Next, we apply an SIR epidemic model to each network layer. The model's key parameters are the transmission probabilities, which represent the daily likelihood of an infected company spreading the infection

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to a susceptible company, and the recovery probabilities, representing the daily likelihood of an infected node becoming recovered. These parameters are estimated using a maximum likelihood approach by fitting the model to past crisis data. We then identify the most vulnerable companies, financial sectors and continents by simulating the spread of financial contagion using the SIR model with these estimated parameters over a given time horizon of interest.

## 2.1 Data

The analysis in this paper relies on the closing daily stock price of 398 companies, spanning from 17/01/2002 to 18/07/2022, representing $N = 5229$ trading days. We emphasise that, in this and all future instances, datasets defined over specified date ranges are considered to include both the start and end dates. The data are sourced from https://finance.yahoo.com/ and the companies are selected such that for each company there are consistent data going back as far as 17/01/2002, covering a sufficient time period before the 2008 financial crisis. The companies are categorized into groups, based on the Orbis<fnref n="1" /> company database. Firstly, they are grouped by the geographical location of their headquarters, resulting in six categories: Africa (2), Asia (77), Europe (115), North America (194), Oceania (9) and South America (1). Secondly, the companies are separated into 13 groups based on their primary economic sector: Finance (47), Oil and gas industry (36), Pharmaceutical industry (36), Automotive industry (35), Airline industry (17), Food industry (23), Mining activities (20), Electricity (17), Software industry (38), Electronics (58), Telecommunications (10), Chemicals (8) and Others (53). The numbers in parentheses indicate the total number of companies in each respective group. More details showing the characteristics of the dataset can be found in [10].

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<footnote n="1">¹Orbis is the world's largest database with a focus on private company financials, ownership and linkages.</footnote>
## 2.2 Network construction

We construct a multiplex network (see Figure 1), in which nodes represent companies and each layer represents a distinct type of connection between them. The network consists of four layers: a tail dependence network layer, a continents layer, a sectors layer and a global layer. The following subsections explain the rationale and methodology used to construct these layers.

### 2.2.1 Tail dependence network layer

To study the likelihood of two companies experiencing extreme losses simultaneously we build complex financial networks based on the concept of tail dependence from Extreme Value Theory (EVT). This process consists of two main steps. We first calculate the strength of the tail dependence between each pair of companies' stock returns. We then use the PMFG (Planar Maximally Filtered Graph) approach to filter and retain the edge information necessary for constructing the network.

#### Tail dependence estimation

Let $\{(-x_{i,t}, -x_{j,t}) : t = 1, 2, \dots, N\}$ be the realisations of the bivariate negative stock return $(X_i, X_j)$, where $x_{i,t}$ is defined as follows:

$$x_{i,t} = \log(p_{i,t}) - \log(p_{i,t-1}). \quad (1)$$

Here $p_{i,t}$ denotes the closing stock price of company $i$ at day $t$ for $1 \le t \le 5229$. Defining $S_i = -1/\log F_i(X_i)$ to be the unit Fréchet variable, corresponding to each marginal distribution function $F_i$ of $X_i$, the upper tail dependence coefficient (upper TDC) $\chi_{i,j}^U$ for each pair $(S_i, S_j)$ and threshold $q \in (0, 1)$ can then be estimated as

$$\hat{\chi}_{i,j}^U = \hat{\chi}_{i,j}^U(q) = \frac{1 - 2q + \hat{C}_{i,j}(q, q)}{1 - q}. \quad (2)$$
In (2), $\hat{C}_{i,j}$ is defined as 277

$$ \hat{C}_{i,j}(u, v) = \frac{1}{N} \sum_{n=1}^{N} 1(r_{i}^{n} \le N - \lfloor N(1-u) \rfloor, r_{j}^{n} \le N - \lfloor N(1-v) \rfloor), \quad 280 $$

and $r_i^n$ and $r_j^n$ are the ranks (the index of the element in an ascending list) of the n-th observations of $S_i$ and $S_j$, respectively. In the case of equal ranks, we assign the average rank to all tied values. The coefficient $\chi_{i,j}^U$ ranges from 0 to 1, quantifying the strength of the tail dependence between $S_i$ and $S_j$. Specifically, $\chi_{i,j}^U = 0$ means that the two variables $S_i$ and $S_j$ are upper tail independent, while $\chi_{i,j}^U > 0$ indicates upper tail dependence. More details can be found in [12]. 284 285 286 287 288 289 290 291 292 293

The analysis throughout the remainder of the paper is based on the estimated upper tail dependence coefficients $\hat{\chi}_{i,j}^U(0.95)$. This threshold choice aligns with the established literature that employs tail dependence for constructing financial networks [13–15]. Moreover, we construct two networks: the first constructed using all available data before the 2008 financial crisis, which includes the data from 17/01/2002 to 30/06/2007 and the second employing all available data before the 2020 financial crisis, i.e., from 17/01/2002 to 29/02/2020. The selection of the $q = 0.95$ value ensures that there are enough data in the upper tail of the distributions before both the 2008 and the 2020 financial crises. 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308

The upper TDC values between each pair of companies i and j are then used to quantify the strength of dependence between the companies in our dataset. These values serve as a basis for constructing multiplex financial networks and implementing the SIR model for financial contagion: a higher TDC between two companies indicates a greater probability of crisis propagation. 309 310 311 312 313 314 315 316 317 318 319 320 321 322
### Planar Maximally Filtered Graph

The Planar Maximally Filtered Graph (PMFG) method, introduced in [16], is designed to filter complex networks by retaining only the most important links while maintaining planarity (the property that a graph can be embedded in a plane without any edges crossing) and ensuring network connectedness. This approach assists with eliminating spurious (weak) connections, highlighting key topological properties such as community structures and reducing computational complexity. In the field of finance, PMFGs have been employed to detect market shifts and community structures [17], to study financial contagion [18] and to analyse financial networks describing correlations (or other dependencies) between financial assets [19]. Furthermore, as shown in [20], PMFGs are useful for reducing the complexity and dimensionality of financial networks, while preserving the original clustering structure. To construct the two PMFG financial networks we employ the procedure described in Algorithm 1. For clarity, we specify that this method is applied only to the tail dependence layer.

1. Start with the graph $G = (V, E)$, where $V$ is the set of nodes (representing companies) and the edge set is $E = \emptyset$.

2. For each pair $S_i$ and $S_j$ ($i < j$) calculate $\hat{\chi}_{i,j}^U(0.95)$ via equation (2) and sort them in a list $l$ from greatest to smallest.

3. Add edge $e_{i,j} = \hat{\chi}_{i,j}^U$ to $E$, corresponding to the index $(i,j)$ of the first element of $l$ if the network $G$ remains planar after addition.

4. Remove $\hat{\chi}_{i,j}^U$ from $l$ and go back to Step 3.

5. Continue the procedure until no more edges can be added.

**Algorithm 1:** Construct PMFG layer

### 2.2.2 Additional layers

Additionally to the tail dependence network (hereafter referred to as the PMFG for brevity) layer, we include layers to represent other known relationships between the companies and to capture alternative channels through which crises may propagate.
## Sector and continents layers 369

The 2008 financial crisis highlighted the critical role of interconnectedness as it quickly spread from the US subprime mortgage market to the broader financial sector. This cascade led to significant losses for financial institutions, while the healthcare and technology sectors demonstrated relative resilience. The crisis affected continents unevenly, highlighting differences in vulnerability and recovery. Similarly, the 2020 crisis, triggered by the COVID-19 pandemic, had sector- and continent-specific effects, benefiting the healthcare sector but hurting travel, tourism, retail and airlines industries. To capture these dynamics, we introduce ‘sectors’ and ‘continents’ layers, where companies are connected if they belong to the same sector or continent, respectively.

## Global layer 387

In the literature there are two types of financial contagion: ‘fundamentals-based’ contagion, known as spillover, where the transmission happens due to real and financial connections and ‘pure contagion’, where crises may propagate due to global effects not explicitly accounted for within the connectivity so far defined. To capture this, we add a complete network, referred to as the ‘global layer’. A schematic representation of our network is shown in Figure 1.

## 2.3 Contagion Model 400

We employ a discrete-time SIR epidemic model defined on the network of C companies to simulate the spread of financial crises. In this model, at any given time step, a company is in one of three states: susceptible (S), infected (I) or recovered (R). Let the functions $S(t)$, $I(t)$ and $R(t) \in \mathbb{Z}_{\ge 0}$ represent the number of companies that are in the states S, I and R, respectively, at time $t$.

The process begins on day $t = 0$, with $M \ge 1$ initially infected companies ($I(0) = M$) and the remaining companies are susceptible ($S(0) = C - M$, $R(0) = 0$).
<figure><img src="image_2.png" /><figcaption>Fig. 1: Schematic diagram of the 4-layer multiplex network described in Section 2.2, comprising a weighted PMFG layer and unweighted continents, sectors and global layers. An inter-layer link is present between each two copies of the same node in different layers and between no other nodes.</figcaption></figure>

At each subsequent day $t = 1, 2, 3, \dots$, an infected company $i$ can infect each susceptible neighbour $j$ on layer $\alpha$ independently with probability $w_{i,j}^{[\alpha]}$. Independently, each infected company recovers with probability $\gamma$ and cannot be reinfected. Consistent with the multiplex nature of our network, the infection or recovery of a company occurs simultaneously across all layers. The process continues until there are no more infected companies.

The per-day transmission probability for each edge $(i, j)$ in layer $\alpha$ ($\alpha = 1, 2, 3, 4$, representing the PMFG, continents, sectors and the global layer) is

$$w_{i,j}^{[\alpha]} = \begin{cases} \hat{\chi}_{i,j}^{U} \times \beta_1, & \alpha = 1, \\ \beta_{\alpha}, & \alpha \in \{2, 3, 4\}, \end{cases} \qquad (3)$$

where $\hat{\chi}_{i,j}^U$ is defined in (2) and $\beta_\alpha$, $1 \le \alpha \le 4$, are parameters to be estimated (see Section 3.4). The definition of ‘infection’ in a financial context is given in Section 3.1.
# 3 Application to financial crises

In this section, we fit the model to the 2008 and the 2020 financial crises. Specifically, in Section 3.1, we define what is meant for a company in the dataset to be ‘financially infected’. Then, in Section 3.2, we describe the ChangeFinder algorithm we use for online CPD. In Section 3.3, we explain the process of incorporating change points into the model. Finally, in Section 3.4 we examine how the model can be used to predict future infections in both financial crises, using recent infection data.

## 3.1 Infection

We define a company as infected whenever the volatility of its stock returns during a specified period surpasses a predefined threshold (indicating stock price instability) and its average stock return for that period is negative (indicating financial loss). In more detail, the volatility of company i for a time horizon $T > 1$ at day t is

$$V_{i,t} = \sqrt{\frac{1}{T} \sum_{j=t-T}^{t-1} (x_{i,j} - \mu_{i,t})^2}, \quad (4)$$

where $x_{i,j}$ is defined in (1) and $\mu_{i,t}$ is the mean stock return over the same period. Thus, company i is considered infected on day t if $V_{i,t} \ge \sigma_i$ and $\mu_{i,t} < 0$, where $\sigma_i$ is the 90% quantile threshold for company i. In the following analysis we set $T = 21$ trading days (one trading month).

Using a rolling window of historical returns over the past 21 days is common in risk analysis [21], particularly for estimating volatility. This method effectively balances the need to capture recent changes in volatility while incorporating enough historical data to provide a stable estimate. Such a balance is crucial for identifying ‘infection’, as longer periods may include stock price fluctuations that no longer influence the market, whereas shorter periods could be more sensitive to noise. The choice of $\sigma_i$ is determined by examining each company’s volatility curve over the entire available

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<figure><img src="image_3.png" /><figcaption>**Fig. 2:** (a) Volatility curve (in blue) and 90% quantile threshold line (in red) for an example company from the dataset. The volatility at day *t* for the period from 07/02/2002 to 29/02/2020 is calculated using the formula in (4) with a rolling window of *T* = 21 trading days; (b) Total number of infected companies per day for the period from 07/02/2002 to 29/02/2020. It can be seen that the highest volatility periods in (a) correspond to the highest peaks in (b).</figcaption></figure>

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timeframe. This value serves as a reference point for assessing the relative volatility of the company's current and future volatility values. However, it is important to note that determining this threshold in real-time, without knowledge of future volatility, is impractical. As a result, the quantile threshold is mainly used as a benchmarking tool to compare and analyse volatility levels across companies in a historical context. An example of a particular company's volatility curve is shown in Figure 2a. The red horizontal line corresponds to a 90% threshold line. We see that with a higher threshold $\sigma_i$, the final, post-2020 'infection period' (high volatility period) would not be captured.

Figure 2b shows the number of infected companies per day, showing that the highest peaks in the numbers of infections are in two periods: from the year 2008 to the year 2010 (in orange), corresponding to the 2008 financial crisis, and from the beginning of 2020 until July 2021 (in purple), corresponding to the COVID-19 crisis. The other infection peaks also correspond to different financial crises, including the early 2000s recession (in blue), the 2012 Cypriot financial crisis (in green) and the 2015–2016 stock market sell-off (in red).
## 3.2 Change point detection

<table><tr><td>Change point detection is a technique for identifying the points at which the probability distribution of a time series changes. It has been employed in statistical analysis for various applications, such as trend analysis [22] and anomaly detection [23]. Algorithms for change point detection are usually classified as ‘online’ or ‘offline’. The ‘offline’ algorithms take into account the whole available data series, where the main goal is to detect all change points. ‘Online’, or real-time, algorithms, on the other hand, operate concurrently with the activity they are monitoring, analysing each data point as it becomes available with the aim of identifying a change point as soon as possible after it occurs, ideally before the next data point arrives. In practice, no change point detection algorithm can actually work in real time, since it must first examine the new data before determining whether a change point happened between the old and new data points. However, such algorithms are still useful in practice, as they can be applied incrementally, allowing for timely updates when new data is processed and facilitating near-real-time adjustments.</td><td>553</td></tr><tr><td>In our study we utilize ChangeFinder [11], an online change point detection algorithm. The ChangeFinder algorithm is particularly well-suited for detecting abrupt shifts or changes in the underlying dynamics of non-stationary time series. This capability is crucial in applications like epidemic modelling, where the transmission rates and recovery rates are likely to evolve over time due to external factors such as interventions, new variants or changes in behaviour. By adapting quickly to these changes, ChangeFinder provides a robust method for real-time anomaly detection, enabling timely responses to unexpected shifts in the spread of an epidemic. Moreover, its ability to continuously update and refine its model of the time series ensures that it can track evolving trends without being overly influenced by outdated data.</td><td>554</td></tr></table>
### 3.2.1 ChangeFinder

Since autoregressive (AR) models constitute the simplest widely-used class of models to represent time series, we begin this section by describing a typical change point detection method based on an AR model.

#### AR Model

Autoregressive (AR) models are an essential concept in time series analysis and forecasting. They are widely used in many fields, including economics and finance [24]. AR models represent the relationship between an observation and previous observations in a time series by expressing the current value as a linear combination of its past values and some random noise.

Let $(z_1, \dots, z_n)$ be a time series with mean 0. Then, the AR model of order $p$, denoted by $AR(p)$ can be defined as

$$z_t = \sum_{i=1}^{p} \alpha_i z_{t-i} + \epsilon_t, \qquad (5)$$

where $\alpha_i$ are the regression parameters, which indicate the degree with which $z_t$ is correlated with $z_{t-i}$, and the $\epsilon_t$ are independent and identically distributed (iid) random variables generated from a Normal distribution with mean 0 and variance $\sigma^2$. Let $\mathbf{x}_{1:n} = (x_1, \dots, x_n)$ be the time series that we observe, where $x_t = z_t + \mu$. That is, the observed data is the $AR(p)$ process shifted by $\mu$, so $\mu$ is the mean of the observed data. Let $\mathbf{x}_{i:j} = (x_i, \dots, x_j)$ represent the segment of the time series spanning from time $i$ to $j$, inclusive. Then, assuming the $AR(p)$ model, the conditional probability density function of $x_t$, given its past, is given by

$$P(x_t | \mathbf{x}_{t-p:t-1}, \theta) = \frac{1}{\sqrt{2\pi}\sigma} \exp \left[ -\frac{(x_t - \omega_t)^2}{2\sigma^2} \right]. \qquad (6)$$
Here, $\omega_t$ is given as follows:

$$ \omega_t = \sum_{i=1}^{p} \alpha_i (x_{t-i} - \mu) + \mu, \qquad (7) $$

and $\theta = (\alpha_1, \alpha_2, \dots, \alpha_p, \mu, \sigma)$ are the model parameters. Then, if $\hat{\omega}_t$ is the estimated value of $\omega_t$ obtained by substituting the estimated model parameters $\hat{\theta} = (\hat{\alpha}_1, \dots, \hat{\alpha}_p, \hat{\mu}, \hat{\sigma})$ into (7), the model-fitting error for $\mathbf{x}_{1:n}$ is given by

$$ I(\mathbf{x}_{1:n}) = \sum_{t=1}^{n} (x_t - \hat{\omega}_t)^2. \qquad (8) $$

The model fitting error $I(\mathbf{x}_{1:t})$ is calculated using maximum likelihood on the data only from time 1 to time $t$, i.e. the parameter $\theta$ is estimated given the observations sequence $\mathbf{x}_{1:t}$. Hence, if $I(\mathbf{x}_{1:t}) + I(\mathbf{x}_{t+1:n})$ is significantly smaller than $I(\mathbf{x}_{1:n})$, then $t$ is a change point. In other words, the procedure assesses if it is significantly better to fit two separate models, one from time 1 to $t$ and the other from time $t+1$ to $n$, than a simpler full model from time 1 to time $n$. In the above formulation we assume that we have only one change point, with the time series being stationary before and after it. However, this assumption can be overly simplistic and unrealistic. The SDAR model, which we discuss in the following section, deals with the situation in which we have multiple change points [11].

## SDAR Model

The SDAR (Sequentially Discounting Auto Regression model learning) algorithm can be used to address the computational problem mentioned above [11]. The SDAR algorithm is employed for adapting and discounting data in an online fashion, utilizing the AR model of order $p$ as follows. For $t > p$, for an estimate $\hat{\theta}_t$ of $\theta$ given $\mathbf{x}_{1:t}$, the SDAR algorithm is employed to calculate in an online manner the value of $\theta$ that maximizes
the following quantity:

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where $r \in (0, 1)$ is a discounting factor. A smaller $r$ indicates a stronger impact from

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700 with non-stationary time series data.

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### ChangeFinder algorithm

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The ChangeFinder algorithm utilizes the SDAR algorithm to detect change points

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in real time. SDAR-based change point detection has been proven to be efficient in

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multiple studies [25–27]. The ChangeFinder algorithm consists of a two-step learning

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process which combines the detection of outliers and the detection of change points in

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a time series. An outlier is a point which indicates a momentary abnormal condition

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(significant increase or decrease), while a change point indicates that the abnormal

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condition does not return to its original state.

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The first learning step of the algorithm uses the SDAR method to calculate an

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outlier score for each data point, which shows how much it deviates from the learned

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outlier. To smooth out short-term fluctuations and reduce the impact of isolated out-

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liers, these scores are averaged over a rolling window of fixed length $T$, producing a

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new time series of smoothed anomaly scores.

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The second stage applies the same procedure to this new time series, effectively

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refining the detection process. By analyzing these refined scores over another sliding

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window, the algorithm identifies points where the data exhibits a sustained change,

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signaling a potential shift in the underlying trend. The detailed step-by-step method

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is described in Algorithm 2.

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$$ \sum_{i=p+1}^{t} (1-r)^{t-i} \log P(x_i|\mathbf{x}_{i-p:i-1}, \theta), \qquad (9) $$
1: Input parameters: order $p$, discounting rate $r$ and rolling window length $T$. 737

2: Receive a new data point $x_t$ at time $t$. 738

3: For each $t \ge p+1$, the outlier score $S(x_t)$ at $x_t$ is calculated, 739
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$$S(x_t) = -\log P(x_t | \mathbf{x}_{t-p:t-1}, \hat{\theta}_{t-1}), \qquad (10) \tag{741}$$

where the values of $\hat{\theta}_{t-1}$ are estimated using the SDAR method with discounting rate $r$, maximizing equation (9), with $P$ given in (6). Here, a higher score $S(x_t)$ indicates that $x_t$ is an outlier with higher probability. 743
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4: For each $t$, a rolling mean of the outlier scores within a time window with fixed length $T$ is computed. In other words, a sequence of moving averages of the outlier scores $y_t$ for $t = T+1, T+2, \dots$ is computed: 746
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$$y_t = \frac{1}{T} \sum_{i=t-T+1}^{t} S(x_i). \qquad (11) \tag{751}$$

Dummy values are chosen for $y_t$, where $t \le T$. This process is needed in order to reduce the influence of isolated outliers contained in the time series $(x_1, \dots, x_t)$. 754
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5: For each $t$, an outlier score $S(y_t)$ at $y_t$ is calculated, 756

$$S(y_t) = -\log P(y_t | \mathbf{y}_{t-p:t-1}, \hat{\theta}_{t-1}), \qquad (12) \tag{757}$$

where $\hat{\theta}_{t-1}$ is estimated using the SDAR method with discounting rate $r$, maximizing equation (9), with $P$ given in (6). 760
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6: A smoothing step is applied to get a sequence of moving averages $a_t$ for $t = 762$
0, 1, 2, ...: 763

$$a_t = \frac{1}{T} \sum_{i=t-T+1}^{t} S(y_i). \qquad (13) \tag{764}$$

Here, $a_t$ is the change point or anomaly score at time $t$. A higher value of $a_t$ indicates a higher probability of $t$ being a change point. 766
767

**Algorithm 2: ChangeFinder algorithm [11]**

## 3.3 Incorporating change points into the model

In this section, we focus on implementing the ChangeFinder algorithm within our model framework. Our goal is to identify and assimilate abrupt shifts or significant events in real-time data and adjust the model's parameters accordingly by considering only the data after a change point is detected and disregarding the data before. By doing so, we allow the model to identify and react to significant shifts, such as market crashes or other economic events, ensuring timely updates and more reliable
predictions. We first show the effectiveness of the ChangeFinder algorithm in detect-
ing change points in the 2008 and 2020 financial crises in an online manner. We
then propose an algorithm for incorporating change points into the model for adap-
tive parameter estimation. We finally demonstrate the effectiveness of change-point
incorporation into the model for predicting future infections.

3.3.1 Real-time change detection in financial crises with
ChangeFinder

Let $I(t)$ denote the total number of infected companies on day $t$ in each of the two crises, estimated using the procedure in 3.1. To detect significant changes in the probability of infection and incorporate them into the model, we examine the time series $\{I(1), I(2), \dots, I(L)\}$, where $L$ is the length of the studied crisis in days. We emphasize that the ChangeFinder algorithm does not use future data when detecting change points. At each time step $1 \le t \le L$, the algorithm receives a new data point $I(t)$ and updates its calculations based only on the observations available up to that point, ensuring that change points are identified in an online, sequential manner. In other words, each anomaly score $a_i$ is calculated based only on the time series $\{I(1), I(2), \dots, I(i)\}$ and does not use the full time series $\{I(1), I(2), \dots, I(L)\}$.

We fit the ChangeFinder algorithm with order $p = 1$, discounting rate $r = 0.5$ and
rolling window length $T = 7$ for the smoothing steps to both the 2008 and the 2020
financial crises. These parameter choices were made to strike a balance between com-
putational efficiency and sensitivity to abrupt changes. In particular, the parameter
$p = 1$ is chosen. Firstly, because it keeps the computational burden low. Secondly,
with $p = 1$, the model adapts quickly to sudden changes: if a sudden spike or drop
in the time series happens, the ChangeFinder algorithm can detect it with minimal
latency. Choosing $T = 7$ strikes a balance between time delay and outlier filtering.
When $T$ is small, outliers and change points can be detected immediately after they
appear, but might be difficult to discriminate from one another. On the other hand, if $T$ is large, it leads to a time delay for detecting change points, but outliers are filtered. Finally, a moderate $r$ value balances between sensitivity to changes and noise in the results. For completeness, Figures B3 and B4 in the Appendix present the anomaly scores for different combinations of the ChangeFinder parameters $r \in \{0.1, 0.5, 0.9\}$ and $T \in \{3, 7, 30\}$, when the algorithm is run on the 2008 and the 2020 financial crises, respectively. It can be seen that the smaller the window size $T$ and the larger the discount rate $r$, the noisier the obtained anomaly scores.

The results for the 2008 and 2020 financial crises, using the chosen parameters, are shown in Figures 3a and 3b. In these figures, anomaly scores and change points are plotted sequentially, reflecting the real-time nature of the algorithm. The top plot in each figure shows the total number of infected companies in black and the solid red vertical lines indicate the days where a high probability of a change point is detected by the ChangeFinder algorithm, or in other words the days with highest anomaly scores, illustrated in the bottom plot in each figure. The other coloured vertical lines correspond to significant events that happened during each crisis, marking key moments that had a substantial impact on the financial system.

During the 2008 financial crisis, some of the most notable events include the freeze in the interbank lending market, which caused liquidity shortages and heightened financial instability. This was followed by the collapse of Bear Stearns in March 2008, a key investment bank that had significant repercussions for global financial markets. The crisis deepened with the bankruptcy of Lehman Brothers in September 2008, which led to a sharp deterioration in market confidence and a global economic downturn. In response to the financial turmoil, the U.S. government implemented several rescue measures, such as the Troubled Asset Relief Program (TARP), which was launched to stabilize the banking sector by purchasing distressed assets. Additionally, the American Recovery and Reinvestment Act (ARRA) of 2009 aimed to stimulate the

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economy through fiscal spending and tax cuts, as well as investments in infrastructure and energy efficiency, seeking to mitigate the effects of the recession.

In the 2020 crisis, the World Health Organization (WHO) declared a global health emergency in March 2020. This was followed by national lockdowns in numerous countries, which significantly impacted businesses. As a result, governments worldwide began implementing stimulus packages to mitigate the economic damage. The introduction of financial support measures helped stabilize markets but did not immediately curb the health crisis. Subsequently, between June and July 2020, many countries began easing their lockdown restrictions, which allowed for some economic recovery. However, this recovery was short-lived, as a resurgence of COVID-19 cases occurred about a month later, signaling the start of a second wave of infections globally.

In both figures it can be seen that most of the time shortly after the mentioned events, a high probability of a change point has been detected. This demonstrates the effectiveness of the ChangeFinder algorithm in identifying significant changes in the data.

### 3.3.2 Incorporating change points for adaptive parameter estimation

In this section we propose a method to adjust the model's parameters dynamically following the detection of a change point. We first construct two separate networks, as detailed in Section 2.2, representing the connections between the companies in the 2008 and 2020 financial crises. The first network is constructed using all available data before the 2008 financial crisis, which includes the data from 17/01/2002 to 30/06/2007. The second network employs all available data before the 2020 financial crisis, i.e., from 17/01/2002 to 29/02/2020. For a detailed comparison of the two networks the reader is referred to [10].
<figure><img src="image_4.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(b)</figcaption></figure>

<caption>**Fig. 3:** Change point detection on the total number of infected companies in the a) 2008 and b) 2020 financial crises, respectively. The anomaly scores and change points are plotted sequentially as they are detected, reflecting the real-time nature of the algorithm. For each crisis the bottom plot illustrates the anomaly scores, as defined in (13). The solid red vertical lines show the highest anomaly scores, while the other coloured vertical lines indicate some of the most important events during both crises.</caption>

To make predictions of the number of infected companies in the future *k* crisis days, given infection data from the past *m* days we utilize a 'sliding window' technique. The

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procedure starts from the first day of the crisis, setting $t = 1$ (data window 1) and we set the initial change point to be $t_c = t$ (the first day of the data window). Then, for each $i \in \{t, \dots, t+m\}$ we calculate the anomaly score $a_i$, using the ChangeFinder algorithm, applied on the time series $\{I(1), I(2), \dots, I(i)\}$. In other words, each anomaly score $a_i$ is calculated based on the time series available up to time $i$ and does not use future data. If $a_i$ is a local maximum, meaning $a_i > a_{i-1}$ and $a_i > a_{i+1}$, within the obtained anomaly scores time series $\{a_t, \dots, a_{t+m}\}$ and $a_i > T$ (the anomaly score is over the threshold $T$), then we classify $i$ as a change point and set $t_c = i$. We disregard all data points before the change point and fit the model to the period from day $t_c$ to day $t+m$ (data window $t$) obtaining maximum likelihood estimates $\hat{\beta}_i$ of the layer transition probabilities $\beta_i$ for $1 \le i \le 4$ and $\hat{\gamma}$ for the recovery rate $\gamma$ (we refer to the Appendix for further details). In other words, instead of fitting the model to all the previous $m$ crisis days, if a change point is detected during the past $m$ days of a crisis, we discard the data before the change point has happened and fit the model exclusively to the days following (and including) the change point. If more than one change point is detected during the past $m$ days, we discard all the data prior to the most recent change point. The step-by-step procedure is described in Algorithm 3.

Next, we simulate $N = 10,000$ realisations of the estimated SIR model for the upcoming $k$ days (from day $m+1$ to day $m+k$, denoted prediction window 1), with the initial data being that from day $m$. We then use the results from the simulations as predictions (see step 5 of Algorithm 3 for more details). We finally 'slide the window' forward by one day and refit the model to the period from day 2 to day $m+1$ of the crisis (data window 2), re-estimating $\hat{\beta}_i$ for $1 \le i \le 4$ and $\hat{\gamma}$ for the new window. We repeat the above steps for each subsequent data window, with the final prediction window covering the period from day $L-k$ to $L$, where $L$ is the length of the crisis in days.
In the following section, we evaluate the model that incorporates change points (we henceforth denote this as the change point model) and compare its performance with that of the model from [10], without incorporating change point detection.

1. Set $t = 1$; set threshold value $\mathcal{T}$.

2. Set $t_c = t$ ($t_c$ is used to record and update the change points' times).

3. For $i \in \{t, \dots, t+m\}$, estimate $\hat{\beta}_i$ (see Step 3(a) or Step 3(b) based on model choice):

(a) **Model without CPD:** Fit the model to the period from day $t$ to day $t+m$ and estimate $\hat{\beta}_i$ ($1 \le i \le 4$) and $\hat{\gamma}$.

(b) **Model with CPD:** For $i \in \{t, \dots, t+m\}$ calculate the anomaly score $a_i$. If $a_i$ is a local maximum within the collected time series of anomaly scores $\{a_t, \dots, a_{t+m}\}$ and $a_i > \mathcal{T}$ (the anomaly score is over the threshold $\mathcal{T}$), then $i$ is classified as a change point. Set $t_c = i$. If more than one change point is detected, set $t_c$ as the most recent change point detected. Fit the model to the period from day $t_c$ (disregard all data points before the change point) to day $t+m$ and estimate $\hat{\beta}_i$ ($1 \le i \le 4$) and $\hat{\gamma}$.

4. Run $N = 10,000$ SIR simulations on the network from day $t+m+1$ to day $t+m+k$ (i.e., predict infections and recoveries in the next $k$ days) using the values of $\hat{\beta}_i$ ($1 \le i \le 4$) and $\hat{\gamma}$ estimated in Step 4. To predict the number of infected companies:

* Count the number of newly infected (recovered, respectively) companies in each simulation and take the mean number of the newly infected (recovered, respectively) companies over all $N$ simulations as a prediction for the number of newly infected (recovered, respectively) companies in the period from day $t+m+1$ to day $t+m+k$ (prediction window $t$).

* Count the total number of infected companies at the end of each simulation and take the mean total number of infected companies over all $N$ simulations as a prediction for the total number of infected companies on the final day of the prediction window $t$.

5. Update $t = t+1$.

6. If $t > L - k - m$ stop. Else go back to Step 2.

**Algorithm 3:** Model predictions for the future $k$ days

## 3.4 Evaluating the change point model: Predicting future infections

In this section we assess the accuracy of the change point model in predicting the number of infected companies in the future $k$ crisis days, given data from the past
1059 *m* days of each crisis. The results show that incorporating change points improves
1060 predictive accuracy, particularly when estimating the number of infected companies.
1061 Unlike the baseline model which performs poorly during and after significant events,
1062 our enhanced approach effectively adapts to these critical moments, yielding better
1063 predictions.
1064

1065
1066
1067 We first compare in Figure 4 how the change point model performs, in comparison
1068 with the model without incorporating the change point algorithm (Algorithm 3 with
1069 step 3a), in predicting the total number of infected companies (top row), number of
1070
1071 newly infected companies (middle row) and the number of newly recovered companies
1072
1073 (bottom row) in the future k = 30 days for different values of m. The outcomes are
1074 derived using Algorithm 3. The results reveal that the model incorporating the change
1075 point detection algorithm (shown in orange) consistently outperforms the model with-
1076
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1078 out the change points (shown in blue), with higher improvement for higher values of
1079 *m*. For completeness, plots for *k* = 10 and *k* = 20 can be found in the Appendix
1080
1081 and show that the results are similar for all choices of *k*. In addition, Figure 5 illus-
1082 trates the model predictions (shown as coloured lines) alongside the observed infection
1083
1084 data (represented by the black curves) at each prediction window. The predictions are
1085 derived from the average of all *N* = 10,000 simulations for the upcoming *k* = 30 days
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1088 of each crisis, using infection data from the previous *m* = 30 days. In the left col-
1089 umn, illustrating the predictions of the model from [10], which does not incorporate
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1091 CPD, it can be seen that significant errors occur shortly after major events, denoted
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1094 by the vertical lines. For instance, in the 2008 crisis, substantial recovery prediction
1095 errors were observed during periods linked with the Lehman Brothers bankruptcy,
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1098 TARP and ARRA. In the 2020 crisis, notable errors were noted shortly after times
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1101 of stimulus packages and lockdown restrictions. Comparing the outcomes of the two
1102 models, we observe a clear improvement in prediction accuracy when CPD is incorpo-
1103 rated, particularly noticeable in the periods following significant events, as indicated
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<figure><img src="image_6.png" /><figcaption>**Fig. 4:** Comparison between the model from [10] and the change point detection model of the absolute difference after *k* = 30 days between predicted and actual total number of infected companies (top row), number of newly infected companies (middle row) and number of newly recovered companies (bottom row) for the 2008 (left column) and the 2020 (right column) financial crises, using the infections data from the previous *m* days.</figcaption></figure>

by the coloured rectangles in Figure 5. In contrast, the model without CPD struggles to adapt to these sudden changes, leading to larger deviations between predicted and actual infections. These results illustrate the importance of incorporating CPD, as it

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<figure><img src="image_7.png" /><figcaption>Fig. 5: Comparison between the model predictions from [10] (left column) and the CPD model (right column). The colored curves represent the predicted average total number of infected companies for each sliding window over the next $k = 30$ days, based on $N = 10,000$ simulation runs. The black lines indicate the actual number of infected companies, as defined in Section 3.1, serving as a benchmark for comparing the model's predictions. The coloured vertical lines indicate important events during each crisis and the coloured rectangles indicate the periods shortly after these events, where the improvements of the CPD model are prominent.</figcaption></figure>

1184
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1188 4 Discussion and conclusion
1189

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1191 In this paper, we incorporated the ChangeFinder algorithm [11], into the model intro-
duced in [10] for change point detection in real time. We achieved this by continuously
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1193 monitoring the data for anomalies, and as soon as a change point was identified, we
discarded the prior data and used only data subsequent to the change point to inform
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model predictions. We investigated the effectiveness of our approach by applying it to two recent financial crises: the 2008 crisis, triggered by the subprime mortgage market collapse, and the 2020 crisis, driven by the COVID-19 pandemic. In each case, we analyzed how well our model estimates future infection risk over a horizon of $k$ days, using data from the preceding $m$ crisis days. Through a comparative analysis, we demonstrated the enhanced predictive performance of the change point model in forecasting the number of infected/recovered companies. The results indicate a consistent improvement over the model without change point detection, highlighting the benefits of integrating change points for a more robust and adaptive predictive model.

While the integration of change points proves advantageous, it is important to acknowledge the potential drawbacks. Firstly, the necessity to choose a threshold (step 1 in Algorithm 3) could impact the algorithm's sensitivity to change points. Moreover, choosing the optimal parameter values depending on the dataset remains an important problem, providing a clear direction for future research. In addition, the ChangeFinder model framework could be extended by incorporating the ARIMA (AutoRegressive Integrated Moving Average) model or a broader model, like the state-space model [28], which could be beneficial for more accurate change point detection.

Another drawback of the model is that it disregards data prior to a detected change point, meaning that potentially valuable information present in older data may be overlooked. While the model's focus on recent data aids in capturing immediate trends, and change points in particular, the omission of historical insights may impact its ability to discern more persistent patterns. To explore this, we applied the CPD model to make continent-, sector-, and company-specific predictions, which were not presented in this paper. Although improvements in these cases were less pronounced, they highlight opportunities for further optimization. A detailed explanation of the accuracy metrics employed to evaluate these predictions can be found in [10]. Future
work could focus on refining these methods to achieve more substantial improvements in these specific contexts.

To address these limitations, a natural direction for future research is introducing an observation weighting approach, which assigns higher weights to more recent observations while still retaining and incorporating older data to a lesser extent. This method aims to balance the need to quickly respond to new data with keeping valuable historical information, leading to higher predictive accuracy. Moreover, in this paper the change point detection was performed on the time series of infections. However, an alternative choice might be performing change point detection on the time series of the estimated model's parameters. In this approach, detecting a change point would indicate a significant shift in a specific model parameter. This method would also give the opportunity to incorporate different change points for the different layers, as opposed to the current approach where detecting a change point in the infection time series leads to reestimating all parameters.

## Acknowledgements

We are grateful for the support of Russell Group Limited for providing the Moody's dataset and for their valuable ideas and discussions throughout the project. We are grateful for access to the University of Nottingham's Augusta HPC service for running simulations used in this study.

## Declarations

## Funding

MB has received funding in the form of an EPSRC CASE studentship EP/XXXXXXX in collaboration with Russell Group Limited.
GS has received financial support from the French National Research Agency 1289
under the grants ANR-23-CE40-0009 (EXSTA project) and ANR-11-LABX-0020-01 1290
(Centre Henri Lebesgue), as well as from the TSE-HEC ACPR Chair and from the 1291
Chair Stress Test, RISK Management and Financial Steering of the Foundation Ecole 1292
Polytechnique. 1293
1294
1295
1296

YvG has received funding from the European Union's Horizon 2020 research 1297
and innovation programme under the Marie Skłodowska-Curie grant agreement No. 1298
777826. 1299
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1302
1303

## Competing interests
---

The authors declare that they have no competing interests. 1305
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## Ethics approval and consent to participate
---

Not applicable 1310
1311
1312

## Consent for publication
---

All authors gave final approval for publication and agreed to be held accountable for 1316
the work performed therein. 1317
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## Data availability
---

The datasets used in this study are available in the GitHub repository, https://github. 1322
com/MBozhidarova98/Integrating_CPD_financial_crisis/tree/main/Datasets. 1323
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## Code availability
---

The codes used in the article are available in the GitHub repository, https://github. 1327
com/MBozhidarova98/Integrating_CPD_financial_crisis/tree/main. 1328
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1334
1335 **Author contribution**

1336
1337 MB collected the data, did the analysis and visualizations shown in the paper and
1338
1339 wrote the manuscript. FB, YvG, GS and RDO did the supervision, reviewed and
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1341 edited the manuscript. All authors read and approved the final manuscript.

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Appendix A Parameter estimation

Once we have determined which companies are infected on each day using the proce-
dure in Section 3.1, we estimate the model’s parameters by maximizing the likelihood
function of the SIR model.

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1528 A.1 Transmission probability

1529
1530 Let us denote the set of infected neighbours in layer α of a susceptible company j at
1531 time t as I_j^α(t). We know that the probability that company i ∈ I_j^α(t) does not infect
1532 company j through layer α at time t+1 is 1 - w_{i,j}^{[α]} (see (3)). Therefore, the probability
1533 that company j does not become infected at time t + 1 is the probability that none of
1534 its infected neighbours in any of the layers 1 ≤ α ≤ 4 at time t infects j at time t + 1:
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1540 P(j does not get infected at time t + 1)

$$
\begin{equation}
\begin{split}
& 1541 \\
& 1542 = \prod_{\alpha=1}^{4} \prod_{i \in I_j^{\alpha}(t)} P(i \text{ does not infect } j \text{ on layer } \alpha \text{ at time } t+1) = \prod_{\alpha=1}^{4} \prod_{i \in I_j^{\alpha}(t)} (1 - w_{i,j}^{[\alpha]}).
\end{split}
\tag{A1}
\end{equation}
$$

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1547 It follows that the probability that company j gets infected at day t + 1 is:

$$
\begin{equation}
P(j \text{ gets infected at time } t+1) = 1 - \prod_{\alpha=1}^{4} \prod_{i \in I_j^{\alpha}(t)} (1 - w_{i,j}^{[\alpha]}). \tag{A2}
\end{equation}
$$

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1556 Let S<sub>t</sub> be the set of susceptible companies at time t. Let SI<sub>t+1</sub> denote the subset of
1557 companies which get infected at time t + 1, so that S<sub>t+1</sub> = S<sub>t</sub> \ SI<sub>t+1</sub>. Therefore the
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likelihood for the observed infections from time t to time t + 1 is:

$$
\begin{equation}
\begin{aligned}
\mathcal{L}_{t,t+1} = & \prod_{j \in \mathrm{SI}_{t+1}} \mathrm{P}(j \text{ gets infected at time } t+1) \\
& \times \prod_{j \in \mathrm{SI}_{t+1}} \mathrm{P}(j \text{ does not get infected at time } t+1).
\end{aligned}
\tag{A3}
\end{equation}
$$

Substituting (A1) and (A2) into (A3) we get:

$$
\mathcal{L}_{t,t+1} = \left\{ \prod_{j \in \mathrm{SI}_{t+1}} \left( 1 - \prod_{\alpha=1}^{4} \prod_{i \in \mathrm{I}_j^{\alpha}(t)} (1 - w_{i,j}^{[\alpha]}) \right) \right\} \left\{ \prod_{j \in \mathrm{SI}_{t+1}} \prod_{\alpha=1}^{4} \prod_{i \in \mathrm{I}_j^{\alpha}(t)} (1 - w_{i,j}^{[\alpha]}) \right\}. \quad (\text{A4})
$$

Suppose we want to estimate the parameters $\beta_i$, $1 \le i \le 4$ for a period of $N$ days from day $d$ to day $d+N-1$. By substituting $w_{i,j}^{[\alpha]}$ with its representation in terms of $\beta_i$, $1 \le i \le 4$, shown in (3), and writing the likelihood of the observations as

$$
\mathcal{L}_{d,d+N-1} = \prod_{t=d}^{d+N-1} \mathcal{L}_{t,t+1}, \quad (A5)
$$

where $\mathcal{L}_{t,t+1} = \mathcal{L}_{t,t+1}(\beta_1, \beta_2, \beta_3, \beta_4)$ is defined in (A4), a maximization of (A5) numerically leads to the maximum likelihood estimate $(\hat{\beta}_1, \hat{\beta}_2, \hat{\beta}_3, \hat{\beta}_4)$ of $(\beta_1, \beta_2, \beta_3, \beta_4)$.

A.2 Recovery probability

Suppose that each company recovers on day *t* with probability *γ*. Let us denote by $\Pi_{t+1}$ and $\mathrm{IR}_{t+1}$ the set of companies which were infected at time *t* and did not recover at time *t* + 1 and the set of companies which were infected at time *t* but recovered at time *t* + 1 respectively. Therefore the likelihood for the recoveries is:

$$
\mathcal{L}_{t,t+1}^{\gamma} = (1 - \gamma)^{|\Pi_{t+1}|} \times \gamma^{|\mathrm{IR}_{t+1}|}. \quad (A6)
$$
<figure><img src="image_8.png" /><figcaption>**Fig. B1:** Comparison between the model from [10] and the change point detection model of the absolute difference after $k = 10$ days between predicted and actual total number of infected companies (top row), number of newly infected companies (middle row) and number of newly recovered companies (bottom row) for the 2008 (left column) and the 2020 (right column) financial crises, using the infections data from the previous $m$ days.</figcaption></figure>

when $T = 3$ or when $r = 0.9$ it can be seen that in all cases there is no change point detected after governments offer stimulus packages (vertical red dotted line) and after the significant increase of COVID-19 cases (vertical green line). Hence, the value of $r = 0.5$ is chosen such that it reduces the noise in the anomaly scores while still allowing for the detection of critical events.
<figure><img src="image_9.png" /><figcaption>1738 **Fig. B2:** Comparison between the model from [10] and the change point detection model of the absolute difference after $k = 20$ days between predicted and actual total number of infected companies (top row), number of newly infected companies (middle row) and number of newly recovered companies (bottom row) for the 2008 (left column) and the 2020 (right column) financial crises, using the infections data from the previous $m$ days.</figcaption></figure>

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<figure><img src="image_10.png" /><figcaption>Fig. B3: The figures illustrate the anomaly scores output, using the ChangeFinder algorithm with different combinations of the discounting rate *r* and the rolling window size *T*. The ChangeFinder algorithm is run on the 2008 daily infections data. The solid red vertical lines illustrate points with high probability of being a change point, whereas the other vertical lines correspond to the significant events that happened during the crisis.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. B4: The figures illustrate the anomaly scores output, using the ChangeFinder algorithm with different combinations of the discounting rate *r* and the rolling window size *T*. The ChangeFinder algorithm is run on the 2020 daily infections data. The solid red vertical lines illustrate points with high probability of being a change point, whereas the other vertical lines correspond to the significant events that happened during the crisis.</figcaption></figure>

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