1. INTRODUCTION
Economic-mathematical modelling has established itself as an exceptionally potent approach for the characterization and examination of intricate socio-economic phenomena and operations. This methodology integrates mathematical models with innovative engineering solutions, thus ingraining it as an indispensable component of economics (Kolosinska and Kolosinskyi, 2013). Knowledge economics, as an abstract concept, necessitates translation into a tangible and concrete manifestation. This transformation is made achievable through the mathematical modelling of its processes as managerial entities (Svarc and Dabic, 2017).
Theoretical exploration and the formulation of mathematical models have yielded favourable results in addressing tangible economic and financial challenges. This includes tasks such as the analysis and prognosis of the flow of budgetary financial resources, the supervision of development, and operational support across varying budgetary tiers, as well as the evaluation of the quality and risk factors associated with the administration of economic systems (Neittaanmäki et al., 2016).
In the wake of the advent of market-oriented economies, global finance, and rapid technological progress, the demand for more sophisticated theoretical underpinning and precise analytical methodologies has surged. Time series models for finance and economics are now leveraging modern signal processing and mathematical tools. These tools encompass time-frequency analysis and the time-scale technique, notably employing the wavelet transform (Tiwari, 2016; Jiang, 2017).
The wavelet transform emerges as a particularly apt choice for scrutinizing economic data due to the nonstationary nature of most economic and financial time series, which exhibit evolving spectral compositions over time.
Prediction stands as a pivotal facet within economics and finance. Long-term prediction is instrumental in governing dynamic economic systems, while short-term prediction seeks to accurately track economic trends. In the examination of economic and financial data for forecasting, the wavelet technique has drawn a lot of interest. Its applications extend to forecasts in areas like crude oil prices, stock market trends, commodity market projections, and more (de Souza, 2010; Tiwari, 2014; Vacha, 2012). However, economic data records frequently contain noise originating from measurement inaccuracies and other sources. Noise within economic and financial systems contributes to heightened uncertainty within financial markets, intensifying the complexity of prediction tasks. To enhance prediction precision, denoising techniques are employed to alleviate data noise. This paper centers its focus on denoising procedures, and harnessing the wavelet transform to purify stock price data for predictive purposes.
Stock price volatility is subject to influence by diverse stochastic elements, thereby generating noisy observations. Through the utilization of wavelet shrinkage denoising and thresholding methods, this paper seeks to formulate a new shrinkage function designed to augment denoising efficacy. The overarching objective is to mitigate the repercussions of noise and elevate the accuracy of stock price predictions.
2. WAVELET TRANSFORM AND SHRINKAGE DENOISING
In the domain of processing and analyzing time series data in economics and finance, the ability to detect meaningful signals amidst noise and interference is paramount. To tackle this challenge, the wavelet transform (WT) has emerged as a valuable tool, leveraging its capacity to pinpoint transient features within signals. Essential properties like compression and sparsity are present in the WT, demonstrating that real-world signals often display sparsity—many smaller coefficients that can be disregarded, while a small number of significant coefficients carry the majority of the signal energy. With an output that rises when the input signal approaches the analysis template and falls when noise levels are higher, the WT essentially performs correlation analysis. This underlying idea supports the idea of wavelet denoising. Importantly, it's imperative to differentiate between wavelet denoising and smoothing, despite some authors interchangeably using these terms. Smoothing is concerned with eliminating high frequencies while preserving low ones, whereas denoising is centered on eliminating noise while retaining the signal, irrespective of its frequency content. Noise energy is distributed across all coefficients in the wavelet domain. Hence, denoising within the wavelet domain is accomplished by applying thresholding to WT coefficients, introducing the notion of wavelet shrinkage. This procedure entails selecting an appropriate threshold value and a thresholding rule to effectively attenuate noise power while retaining essential signal characteristics. Diverse wavelet thresholding strategies have been shown to exhibit near-optimal qualities from a minimax perspective in several research concentrating on signal denoising by wavelet shrinkage (Donoho, 1998).
The WT is widely used in non-stationary time series analysis to extract data in both the frequency and time domains. It can be viewed as a multi-scale variant of the specialized Fourier transform that dissects signals into smaller and altered forms of the original "mother" wavelet. The convolution of a time series, represented as x(t), and a wavelet function, represented as w(t), is the formal definition of the continuous wavelet transform (CWT) (Gomes,2015)
(1)
The translational parameter is denoted by d
, the scale parameter is represented by a= 
,
and the complex conjugate of ψ(t),
is denoted by ψ*, where "s" and "k" are integers. At
the coordinates (
, 
) on the plane of time
scale, the CWT of x(t) yields a numerical result. This outcome displays the
relationship at that specific time-scale point between x(t) and ψ*(t). The
discrete counterpart of equation (1) is known as the discrete wavelet transform
(DWT):
(2)
The signal is broken down by DWT into components with different scales that correspond to successive frequencies. DWT plays a pivotal role in the multi-resolution approximation approach for analyzing signals across multiple scales or frequency bands.
In application, multi-resolution analysis starts with two-channel filter banks made up of low-pass and high-pass filters, and each filter bank is sampled half as fast as the preceding frequency (1/2 down-sampling). The length of the data determines how many rounds there will be in this decomposition. This down-sampling technique sustains a constant scaling parameter of 1/2 across successive wavelet transformations (Li and Kuo, 2008), thereby streamlining the computational implementation. DWT has seen extensive development and application in signal analysis across diverse domains.
By creating a suitable shrinkage (thresholding) function, DWT is used in this work to remove noise from stock price data. Risk Shrink, a useful spatially adaptable technique that trims empirical wavelet coefficients, is a part of groundbreaking research on wavelet shrinkage denoising (Donoho, 1994). Another well-known technique is the asymptotically minimax approach, which, when applied to curve estimation in the presence of noisy data, moves empirical wavelet coefficients toward the origin. Furthermore, Donoho and Johnstone (1995) describe Sure Shrink, as a technique for reducing noise by thresholding empirical wavelet coefficients. All these techniques are combined under the name "Wave Shrink," which is based on the idea of eliminating noise by setting wavelet coefficients to zero. A widely accepted technique for removing noise from distorted signals is called Wave Shrink.
Numerous scholars have employed wavelets in diverse applications in economics and finance, deploying DWT to dissect economic and financial data. This opened the door for wavelet analysis to be used in empirical economics and finance. The quasi-periodic components of signals are extracted using the Ensemble-Empirical-Model Decomposition (EEMD) and ensemble version of Empirical-Model-Decomposition (EMD) methods (Huang, 1998) in order to construct features. By using these nonparametrically produced components as inputs for classification models, human bias in feature generation can be minimized. Many fields have successfully used EEMD and DWT, particularly when dealing with non-stationary time series.
Non-stationary time series and highly oscillatory, connected to financial instruments like stock prices and market indices have been forecasted in a number of studies using EEMD and DWT.
Wavelet analysis is well known for its ability to examine the frequency content of analyzed processes with high joint time–frequency resolution and for estimating economic and financial time series using particular wavelet–based techniques. This method, which is based on an endogenously variable time window, allows for accurate identification of events that impact economic fluctuations, effective computational analysis, and the examination of economic relationships across various time horizons. In conclusion, wavelet analysis is crucial for economic forecasting even beyond its use in signal denoising. Conventional DWT has an advantageous decorrelation property, which suggests that it could be useful for financial and economic process forecasting. Surprisingly, the choice of analysis scale has a significant impact on the outcomes of forecasting methods used for financial and real economic variables. Wavelet transforms (CWT and DWT) simplify the analysis of simpler univariate and multivariate processes, which makes it easier to apply customized forecasting methods.
In the context of this analysis, we
introduce data (time series) denoted as 
, defined as follows:
(3)
represents normalized
time in this equation, f(·) is a deterministic function with potential
complexity and spatial inhomogeneity, The Gaussian noise standard deviation is
represented by the variable{
}, with 
N(0,1). The objective is
to minimize the 
risk when estimating
f(·) (Donoho, 1994; 1995):
(4)
The estimated sample value of f
is denoted by 
. The mean square error (MSE) is represented by the risk. A well-liked tool
for estimating the function f(·) is WaveShrink. It achieves nearly optimal
minimax risk within a logarithmic factor of n across a broad range of
smoothness classes and loss functions, including risk, and has broad
asymptotic near-optimality properties. The following steps are involved in the
WaveShrink method is done in three steps: a) Wavelet coefficient thresholding,
which involves shrinking empirical wavelet coefficients to zero, is done in three
steps: (b) forward wavelet transform of observed data, which turns data y into
the wavelet domain; (c) the reduced coefficients' inverse wavelet transform,
which returns the shrunken coefficients to the data domain.
The efficiency of shrinking empirical wavelet coefficients depends on the sparsity of the true coefficients of f(·).
This translates to practice as most coefficients (the non-important ones) have little effect on the functional form of f(), while a small number (the important ones) have a significant effect. A shrinkage function, sometimes referred to as a thresholding function, needs to have two characteristics to be accepted: a) It must retain large and significant coefficients and b) It must eliminate small and insignificant ones.
Hard shrinkage and soft shrinkage are the two basic shrinkage functions that Wave Shrink uses. Important coefficients are retained by the hard shrinkage function, but non-important coefficients are set to zero if their absolute values fall below a predetermined threshold ζ:
(5)
Here, ζ denotes the threshold value, with ζ ∈ [0, ∞).
Expanding upon the hard shrinkage function is the soft shrinkage function. Non-essential coefficients are set to zero and crucial coefficients that are not zero are reduced to zero if their absolute values fall below the threshold.:
(6)
There are advantages and disadvantages to both soft and hard shrinking. Large coefficients' shrinkage may cause a larger bias since the soft shrinkage is continuous but its derivative is discontinuous. Hard thresholding estimates, on the other hand, may have a higher variance and be unstable due to discontinuities in the shrinkage function (Gao 1998). The Wave Shrink denoising technique utilizes Breiman's (1995) non-negative garrote shrinkage function to solve the drawbacks of hard and soft shrinkages. The following is the definition of the non-negative garrote shrinkage function:
(7)
Several conclusions can be drawn
from the three shrinkage functions discussed: At the threshold 
, the hard functional
shrinkage shows a discontinuity. The soft shrinkage function maintains
consistency across its entire range. Since it is continuous, the non-negative
garrote shrinkage function is better than hard shrinkage. Moreover, the
positive garrote shrinkage function reaches the identity line as |x| increases
(just like hard shrinkage does), resulting in a lower bias for high
coefficients than soft shrinkage. These findings suggest that a good compromise
between hard and soft shrinkage functions is the non-negative garrote shrinkage
function.
It is worth noting, however, that
all three functions share the property of having a single threshold. (Gao,
1997) proposed a more general shrinkage function known as the firm (or
semisoft) shrinkage function, which includes two thresholds (
and 
):Top of Form
(8)
 |
The sign function is represented by "sgn()" here. When x is close to the lower threshold
, the firm shrinkage 
behaves similarly to the
soft shrinkage 
. In contrast, for x
values greater than the upper limit , 
equates to 
, which is equivalent to
x. Notably, firm shrinkage with equal to corresponds to hard
shrinkage, whereas firm shrinkage with approaching infinity
mimics soft shrinkage. As a result, the firm shrinkage function extends and
generalizes the hard and soft shrinkage functions found in Wave-Shrink.
Semisoft shrinkage can outperform both hard and soft shrinkage methods by
carefully selecting thresholds (, ), incorporating the
benefits of both while avoiding their drawbacks (Gao 1998). The requirement for
two thresholds, however, is a major drawback of semisoft shrinkage and can make
threshold selection more difficult as well as increase computational complexity
in adaptive threshold selection processes.
The waveforms of the hard (ζ =
3.41), soft (ζ = 3.41), positive garrote (ζ = 3.41), and semisoft ( = 2.341, = 7.264) shrinkage
functions are shown in Figure 1. Although thresholding functions with improved
performance, such as the non-negative garrote and semisoft, which strike a
balance between hard and soft functions and have advantages over both, can
occasionally lack higher-order differentiability and have fixed structures that
limit their adaptability and flexibility. As a result, alternative thresholding
functions with continuous higher-order derivatives must be investigated.
3. DEVELOPMENT OF NOVEL NONLINEAR THRESHOLDING FUNCTIONS
To streamline the process of threshold selection and alleviate computational complexity, we aim to devise a novel shrinkage function that relies on a single threshold parameter, denoted as ζ. Simultaneously, this designed t thresholding function should endeavor to address the limitations observed in both the hard and soft shrinkage functions to the greatest extent possible.
Figure 1: Waveform Characteristics of the Shrinkage Functions.
The development of the novel thresholding function adheres to foundational principles. To address shortcomings in existing shrinkage functions, the proposed function prioritizes continuous derivatives, ensuring adaptability for gradient-based algorithms. Additionally, it maintains wavelet coefficients unchanged when their absolute values exceed the threshold, approximating a linear function (y=x). Crucially, the function zeroes coefficients below a given threshold (ζ), akin to y=0. These principles aim to minimize estimation variance and bias, overcoming limitations in conventional hard and soft shrinkage methods. The desired properties include yielding y=0 for coefficients within [-ζ, ζ], and y=x for coefficients beyond this range. These specifications guide the search for a function aligning with these criteria, ensuring the theoretical efficacy of the proposed shrinkage function.
The arctangent function is widely
utilized for improved approximation when approximating a function is close to
the origin. The arctangent function is the inverse of the tangent function,
denoted as 
. It has the domain of
real numbers and the range of 
. The horizontal
asymptotes of the arctangent function are 
and 
. It is a one-to-one
relationship. Figure 2 depicts the arctangent function's waveform.
Figure 2: The Arctangent Function Waveform
When the x value is very small, the
arctangent function should be passes through the origin, i.e., 
, as shown in Figure 2.
When the x value moves away from
the origin, 
increases. These traits
match the intended criteria of the thresholding function. As a result,
arctan(x) can be used as the starting point for constructing the new shrinkage
function with continuous derivatives. To rase the adaptability and functionality
of the suggested shrinkage (thresholding) function, three shape-tuning
parameters are included:
(9)
In this context, determining real numbers α, β, and k is essential. A nonzero positive integer for k ensures the thresholding function's odd symmetry. To accommodate a diverse signal range, the suggested shrinkage function is not only continuous but possesses higher-order derivatives. Examining differentiability, the continuity and equality of left and right derivatives at x=ζ are both necessary and sufficient conditions for the function's differentiability concerning the threshold variable.
3.1 The Proposed Shrinkage Function's Partial Derivative
The 
is the partial
derivative of the presented shrinkage function.
(10)
This derivative ensures continuity and is necessary for the proposed shrinkage function to be smooth. The parameter β can be calculated as follows:
β =
(11)
3.2 The Proposed Shrinkage Function's Second-Order Partial Derivative
 |
The presented shrinkage function's second-order partial derivative is as follows:
(12)
The parameter k can be determined as:
(13)
3.3 The Final Shrinkage Function:
Taking into consideration the above equations and parameters, the differentiable shrinkage function can be expressed as:
(14)
3.4 Parameter α
Optimization:
Prior to putting the shrinking function into use, the value of the parameter must be predetermined. The formula (14) can be simplified to determine the optimal value of
: 
(15)
Upon analyzing the formula (15), two critical results emerge:
(a) When 
, the shrinkage function should closely resemble hard thresholding,
making the value of α as small as possible. Notably, when α=1,
the value 
reaches a minimum
(specifically, =0.57). This implies that
the priested shrinkage function is similar to non-negative garrote shrinkage
for significant wavelet coefficients.
(b) When 
, The non-important
wavelet coefficients should be set to zero by the shrinkage function. in order
to reduce noise. α should be as large as possible in this case.
To reconcile these contradictory results,
the optimal α value should take into account both conditions
thoroughly. Furthermore, because it balances estimate variance and bias, the risk, or (MSE) will be
used as a criterion for determining the optimal α value. The
optimal value and the formulas for bias, variance, and risk of the suggested
shrinkage estimate for a normal random variable are discussed in the section
that follows.
RISK ANALYSIS AND
DETERMINATION OF OPTIMAL THRESHOLDS
For convenience, we represent the
proposed shrinkage function as 
Let X follow a Gaussian
distribution, X~N(θ, 1). We determine the mean, variance, and risk functions for the
shrinkage estimate of θ under the shrinkage function 
(·) and threshold , as follows (Gao 1998):
Mean:
,Variance:
Risk:
((16)
Here, 
represents the square
bias. We can further simplify Equation (16) to: 
− 2θ ·
According to Equation (16), the bias
component contributes more to the than the variance
component. As a result, minimising bias by selecting an optimal α value
reduces the risk function. The 
risk function can
nevertheless accomplish an ideal trade-off between bias and variance even when
the variance is not at its lowest in this situation. To find the parameter
value that minimizes the risk, the risk functions for
various values of can be computed and compared. For example, we could take = 1,
2, 3, 5, 7 and set the threshold to = 3.41. Figure 3 depicts the resulting
waveforms of the risk functions for the
presented shrinkage function.
Figure 3: shows the risk profiles of the
presented shrinkage function at variate α values.
Figure 4: depicts the risk functions for the
hard, soft, and proposed functions
In Figure 4. we have included for hard shrinkage ( = 3.41), soft shrinkage
( = 3.41) have been
included, and the study presented shrinkage functions ( = 4.31) to facilitate
comparison.
This implies that the proposed
function with α = 1 offers an advantage in the VisuShrink method, which
employs the universal threshold. According to the VisuShrink method, the
threshold is given by:
(17)
 |
Where σ is the normal white noise standard deviation, and N is the total number of wavelet coefficients, which is typically equal to the number of samples in the noisy data. The universal threshold helps avoid unwanted artifacts in the reconstructed data by setting small coefficients to zero. The universal threshold, on the other hand, is dependent on knowing the noise intensity, which is frequently unknown in practical scenarios. To address this, the intensity σ of the noise from the data itself can be estimated. The finest scale empirical wavelet coefficients yield the following estimate for σ:
(18)
In this case, 
denotes the noisy
wavelet coefficients, and 
denotes the number of
noisy data points. In addition to the VisuShrink method, we consider the
SureShrink procedure, which uses empirical wavelet coefficient thresholding to
suppress noise (Donoho,1995). The adaptive thresholding aims to minimise the Stein
unbiased estimate of risk (Sure) for threshold estimates. This execution's
computational effort is proportional to N·log(N), where N is the sample size.
5. APPLICATION METHODOLOGY
We a dataset comprising closing prices of the Shanghai Composite Index (SCI) was utilized and sourced from the Shanghai Stock Exchange (SSE) for our numerical analysis. This dataset encompasses a total of 5789 trading days, equivalent to approximately 283 trading months, spanning from Jan. 4, 2000, to Aug. 28, 2023. The raw data series illustrating the closing prices for each trading day are presented in Figure 5.
Figure 5: Plots Depicting the Original Daily Stock Price Data Series.
