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# New Algorithm for Weak Changes Detection with Application to Financial Data

Youssef Salman, Anis Hoayek, Mireille Batton-Hubert

► To cite this version:

Youssef Salman, Anis Hoayek, Mireille Batton-Hubert. New Algorithm for Weak Changes Detection with Application to Financial Data. Sustainable Progress Through Innovation in Data Science, Health, Environment, and Social Systems, Jul 2025, On Line, France. hal-05263325

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https://hal.science/hal-05263325v1

Submitted on 16 Sep 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# New algorithm for weak changes detection with
application to financial data

Youssef SALMAN* ¹ Anis HOAYEK ¹ Mireille BATTON-HUBERT ¹

¹Henri Fayol Institute, Mines School Engineering, Saint-Étienne, France

## Introduction

Detecting weak changes in time series is crucial in many fields, particularly in finance and economics, where subtle structural shifts may precede major market events. Traditional methods often fail to identify these small yet impactful changes. Our work proposes a novel statistical procedure capable of detecting such weak changes, even when masked by noise, and precisely estimating their occurrence times.

## Piece-wise stationary CHARN models

Let $d, p, k, n \in \mathbb{N}$ and $k \ll n$. Assume the observations $X_1, \dots, X_n$ issued from the following piece-wise stationary CHARN model (see, e.g., [1])

$$X_t = T(\rho_0 + \gamma \odot \omega(t); \mathbf{X}_{t-1}) + V(\mathbf{X}_{t-1})\varepsilon_t, t \in \mathbb{Z}, \quad (1)$$

with

$$X_t = Y_{t,j} = T(\rho_0 + \gamma_j \omega_j(t); \mathbf{X}_{t-1,j}) + V(\mathbf{X}_{t-1,j})\varepsilon_t, \quad \tau_{j-1} \le t < \tau_j, \quad (2)$$

where for $j = 1, \dots, k$, $(Y_{t,j})_{t \in \mathbb{Z}}$ is a stationary and ergodic process; $\rho_0 \in \mathbb{R}^p$, $T(\rho_0, \cdot)$ and $V(\cdot)$ are real-valued functions with $\inf_{x \in \mathbb{R}^d} V(x) > 0$; the $\tau_j, j = 0, \dots, k+1$, are potential instants of changes with $\tau_0 = 1$ and $\tau_{k+1} = n+1$; for $j = 1, \dots, k$, $\mathbf{X}_{t,j} = (Y_{t,j}, \dots, Y_{t-d+1,j})^\top$, $\mathbf{X}_{\tau_{j-1}+\ell} = \mathbf{X}_{\tau_{j-1}+\ell,j}$, $\ell = 0, \dots, d-1$ and for $t \in [\tau_{j-1} + d - 1, \tau_j)$, $\mathbf{X}_t = (X_t, \dots, X_{t-d+1})^\top$; for $j, \ell = 1, \dots, k, j \ne \ell$, the processes $(Y_{t,j})_{t \in \mathbb{Z}}$ and $(Y_{t,\ell})_{t \in \mathbb{Z}}$ are mutually independent; $(\varepsilon_t)_{t \in \mathbb{Z}}$ is standard white noise with density $f$. $\gamma = (\gamma_1^\top, \dots, \gamma_{k+1}^\top)^\top$, $\gamma_j \in \mathbb{R}^p, j = 1, \dots, k+1$; $\omega(t) = (\mathbf{1}_{[\tau_0, \tau_1)}(t), \mathbf{1}_{[\tau_1, \tau_2)}(t), \dots, \mathbf{1}_{[\tau_{k-1}, \tau_k)}(t), [-35]\mathbf{1}_{[\tau_k, \tau_{k+1})}(t))^\top = (\omega_1(t), \dots, \omega_{k+1}(t)) \in \{0, 1\}^{k+1}$; for $\gamma = (\gamma_1^\top, \dots, \gamma_{k+1}^\top)^\top$ and $\omega(t) = (\omega_1(t), \dots, \omega_{k+1}(t))^\top$, $\gamma \odot \omega(t)$ stands for $\gamma \odot \omega(t) = \gamma_1 \omega_1(t) + \dots + \gamma_{k+1} \omega_{k+1}(t) \in \mathbb{R}^p$, and $\gamma_i \omega_i = (\gamma_{i,1} \omega_i, \dots, \gamma_{i,p} \omega_i) \in \mathbb{R}^p$.

For $\gamma_0 \in \mathbb{R}^{p(k+1)}$ and $\beta \in \mathbb{R}^{p(k+1)}$ depending on the $\tau_j$s, we construct a likelihood ratio test for testing

$$H_0: \gamma = \gamma_0 \quad \text{against} \quad H_\beta^{(n)}: \gamma = \gamma_n = \gamma_0 + \frac{\beta}{\sqrt{n}}. \quad (3)$$

We studied the case where the parameters of the models are known and unknown, you can check [3] for more info. Here, we present the final results needed to use in case where the parameters are unknown.

## Explicit expression of the power of the test

Under several assumptions (check [3]), for any given $\beta \in \mathbb{R}^{p(k+1)}$, the asymptotic power of the test constructed is $P_{k,\tau_k} = 1 - \Phi(z_\alpha - \vartheta(\rho_0, \gamma_0, \beta))$, where $z_\alpha$ is the $(1-\alpha)$-quantile of the standard Gaussian distribution with cumulative distribution function $\Phi$ and $\vartheta$ is a known function with known form (check [3]). In practice, $\hat{\vartheta}_n(\rho_0, \gamma_0, \beta)$ can be taken as a natural estimator of $\hat{\eta}_n^{\frac{1}{2}}(\rho_0, \gamma_0, \beta)$ with

$$\hat{\eta}_n(\rho_0, \gamma_0, \beta) = \sum_{j=1}^{k+1} \hat{\alpha}_j \sum_{1 \le h \le m \le p} \beta_{j,h} \beta_{j,m} \hat{\eta}_{j,2}^{(h,m)}(\rho_0, \gamma_0),$$

and for $j = 1, \dots, k+1$, $\hat{\alpha}_j$ is an estimator of $\alpha_j = \lim_{n \to +\infty} n_j(n)/n$ and

$$\hat{\eta}_{j,2}^{(h,m)}(\rho_0, \gamma_0) = I(f) \int_{\mathbb{R}^d} \frac{1}{V^2(x)} \frac{\partial T}{\partial \gamma_{j,h}}(\rho_0, \gamma_0, x) \frac{\partial T}{\partial \gamma_{j,m}}(\rho_0, \gamma_0, x) d\hat{F}_j(x),$$

where $\hat{F}_j$ is the empirical distribution function of the observations with indices in $[\tau_{j-1}, \tau_j)$.

## Automatic algorithm for detecting the weak changes [2]

Let $\zeta \in (0, 1)$ and $X_1, X_2, \dots, X_m, (m \ll n)$, the $m$ first stationary observations. A crucial point to mention is that, in practice, $m$ will be small than that considered in [3].

Our procedure for detecting weak changes in the time series $X_1, X_2, \dots, X_n$ and estimating their locations is described in the following algorithm.

### Location 1:

Put $t = 1$

($S_1$): Consider the two intervals $I_1$ and $I_2$ that contains respectively the observations $X_{1,1}, \dots, X_{m+t-1}$ and $X_1, \dots, X_{m+t}$. So that the difference between the two intervals considered is the single observation $X_{m+t}$ which is under testing.

($S_1'$): Adjust model (1) to $I_1$ and $I_2$. Then, apply the testing procedure presented in [?].

If $|P_{1,t} - P_{0,\tau_0}| > \zeta$,

Replace $X_{m+t}$ with $X_{m+t}$ in $I_2$, with $t+1 \le \iota \le j$, $j \ll m$, and Repeat ($S_1'$) with the updated $I_2$

If $|P_{1,\iota} - P_{0,\tau_0}| > \zeta$,

The first change location is estimated on $\tau_1 = m + t$.

Then, Go to **Location 2**.

Else

A False Alarm is detected.

Remove $X_{m+t}$ from the sample, Do $t = t+1$ and Go to ($S_1$).

Else

Do $t = t+1$ and Go to ($S_1$).

### Location 2:

Consider the next $m$ observations to $X_{\tau_1}$: $X_{\tau_1+1}, \dots, X_{\tau_1+m}$

Put $t = 1$ and Do

($S_2$): Consider the two intervals $I_1$ and $I_2$ that contains respectively the observations $X_{\tau_1}, \dots, X_{\tau_1+m+t-1}$ and $X_{\tau_1}, \dots, X_{\tau_1+m+t}$. So that the difference between the two intervals considered is the single observation $X_{\tau_1+m+t}$ which is under testing.

($S_2'$): Adjust model (1) to $I_1$ and $I_2$. Then, apply the testing procedure presented in [?].

If $|P_{1,t} - P_{0,\tau_0}| > \zeta$,

Replace $X_{\tau_1+m+t}$ with $X_{\tau_1+m+t}$ in $I_2$, with $t+1 \le \iota \le j$, $j \ll m$, and Repeat ($S_2'$) with the updated $I_2$

If $|P_{1,\iota} - P_{0,\tau_0}| > \zeta$,

The second change location is estimated on $\tau_2 = \tau_1 + m + t$

Then, Go to **Location 3**.

Else

A False Alarm is detected.

Remove $X_{\tau_1+m+t}$ from the sample, Do $t = t+1$ and Go to ($S_2$).

Else

Do $t = t+1$ and Go to ($S_2$).

### Location i:

We already estimated the $(i-1)^{th}$ change location $\tau_{i-1}$ in step $i-1$

Consider the next $m$ observations to $X_{\tau_{i-1}}$: $X_{\tau_{i-1}+1}, \dots, X_{\tau_{i-1}+m}$

Put $t = 1$ and Do

($S_i$): Consider the two intervals $I_1$ and $I_2$ that contains respectively the observations $X_{\tau_{i-1}}, \dots, X_{\tau_{i-1}+m+t-1}$ and $X_{\tau_{i-1}}, \dots, X_{\tau_{i-1}+m+t}$. So that the difference between the two intervals considered is the single observation under testing.

($S_i'$): Adjust model (1) to $I_1$ and $I_2$. Then, apply the testing procedure presented in [?].

If $|P_{1,t} - P_{0,\tau_0}| > \zeta$,

Replace $X_{\tau_{i-1}+m+t}$ with $X_{\tau_{i-1}+m+t}$ in $I_2$, with $t+1 \le \iota \le j$, $j \ll m$, and Repeat ($S_i'$) with the updated $I_2$

If $|P_{1,\iota} - P_{0,\tau_0}| > \zeta$,

The $i^{th}$ change location is estimated on $\tau_i = \tau_{i-1} + m + t$.

Then, Go to **Location i + 1**.

Else

A False Alarm is detected.
Remove $X_{\tau_{i-1}+m+t}$ from the sample, Do $t = t+1$ and Go to ($S_i$).
Else
Do $t = t+1$ and Go to ($S_i$).

## FSTE 100 index

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## Significant Events During the Detected Periods

* February, 2007, and July, 2007, reflects early signs of market volatility linked to the subprime crisis

* May, 2006: likely reflects global market concerns about inflation and interest rates, which triggered a broader sell-off.

* January, 2008, March, 2008, and September, 2008, providing a more detailed representation of market turbulence during the 2008 financial crisis.

## S&P 500 index

<figure><img src="image_4.png" /></figure>

## Significant Events During the Detected Periods

* February 1993, a change point which could be linked to the inauguration of President Bill Clinton and the anticipation surrounding his economic policies aimed at reducing the federal deficit.

* Mars 1994, which we linked to the U.S. lifting of the trade embargo on Vietnam

* September 1994, identified by the new algorithm, corresponds to the bond market crisis, also known as the "Bond Market Massacre" of 1994

* April 1997, which likely reflects the market's reaction to early concerns about inflation and potential interest rate hikes

* Mars 1996, a subtle change linked to the Federal Reserve's decision to hold interest rates steady after prior hikes

* June and November 1998, which were connected to market reactions to the Long-Term Capital Management (LTCM) crisis and the Federal Reserve's rescue operation.

* August 1998, coinciding with the start of the Russian financial crisis

## Highlights of the Algorithm

Once implemented, the algorithm runs efficiently and provides fast detection of weak changes in the data. Each estimated change point is accompanied by the corresponding test power, offering immediate insight into detection strength. This makes the method practical and effective for real-time or large-scale time series analysis.

## References

[1] Wolfgang Härdle, Alexandre Tsybakov, and Lijian Yang. Nonparametric vector autoregression. *Journal of Statistical Planning and Inference*, 68(2):221-245, 1998.

[2] Youssef Salman, Anis Hoayek, and Mireille Batton-Hubert. New algorithm for detecting weak changes in the mean in a class of charn models with application to welding electrical signals. *Engineering Proceedings*, 68(1):42, 2024.

[3] Youssef Salman, Joseph Ngatchou-Wandji, and Zaher Khraibani. Testing a class of piece-wise charn models with application to change-point study. *Mathematics*, 12(13):2092, 2024.

[4] Claude E. Shannon. A mathematical theory of communication. *Bell System Technical Journal*, 27(3):379-423, 1948.