# Exploring the Effect of Sequence Smoothness on Machine Learning Accuracy

Cangqing Wang${}^{1}*$ and Hoc T Quach2 ${}^{1}*$Cangqing Wang be with Boston
University, MA 02215, USA<EMAIL_ADDRESS>T Quach is an independent
researcher. Correspondence to Hoc T Quach via email<EMAIL_ADDRESS>
###### Abstract

Time series data, amassed through regular intervals over time, has become
abundant with the rapid evolution of science and technology. Analyzing these
time series unveils inherent dynamics, facilitating a nuanced understanding of
diverse social phenomena. Moreover, such analyses empower future predictions,
establishing a scientific foundation for informed decision-making and
regulatory measures. Consequently, the importance of enhancing the accuracy of
time series predictions cannot be overstated.

Addressing the challenges posed by large-scale, non-linear, and non-smooth
time-series data, numerous machine learning-based time-series prediction
methods have emerged and found extensive application across various
industries. Long Short-Term Memory (LSTM) networks, characterized by their
specialized architecture facilitating long-term memory retention, are
particularly well-suited for modeling and predicting real-world time-series
data. While significant research efforts have concentrated on optimizing the
LSTM model structure to enhance prediction accuracy, limited attention has
been given to the preprocessing inadequacies in machine learning for time-
series data.

This paper adopts a perspective grounded in statistical theory and machine
learning methods to explore the impact of sequence smoothness on machine
learning accuracy through data preprocessing. Utilizing Python software, the
study simulates eight non-smooth series, including random wandering with
smooth + AR(1) sequence, random wandering with + MA(2) sequence, random
wandering with + smooth ARMA(1, 1) sequence, random wandering with + smooth
SARMA(1, 10, 1) sequence, linear trending smooth sequence, linear trending
with + smooth AR(2) sequence, quadratic trending smooth sequence, and linear
trending with + smooth AR(1, 10, 1) sequence, each of length 300.

Subsequently, employing the time series split cross-validation method, the
original data and differenced data are segmented into multiple training and
testing sets for model training and prediction. The study calculates the
prediction accuracy of the original series and the inverse prediction accuracy
of the differenced series. Comparisons of LSTM learning effects before and
after differential smoothing reveal that preprocessing the original series
with differential smoothing significantly improves the learning accuracy of
LSTM when modeling non-stationary time series. Lastly, to validate the
generality of the findings, daily closing price data of SSE and SZSE indices
from December 31, 2012, to December 31, 2022, are utilized. To optimize the
original LSTM from a data preprocessing perspective, a novel differential-LSTM
algorithm, combining differential smoothing with LSTM, is proposed. This
approach contributes to advancing the understanding and application of machine
learning in optimizing time-series prediction accuracy.

###### Index Terms:

Sequence smoothing; machine learning accuracy; LSTM; time series prediction

##  I Introduction

Time series analysis is crucial, in industries and everyday life providing
insights into trends in historical data. This method helps us understand
phenomena and make predictions about the future guiding decision making
processes. The accuracy of these predictions greatly influences how decisions
are made leading experts in fields to focus on improving time series
forecasting precision.

With advancements in computer technology there are now machine learning based
techniques for time series forecasting. These new methods address the
limitations of forecasting approaches when dealing with real world problems.
Machine learning algorithms can handle amounts of data and nonlinear patterns
by adjusting weights based on desired outcomes. Unlike methods that require
data machine learning can predict changes, in sequences that are not constant.
The concept of stationarity, which refers to the consistency of properties
over time regardless of the starting and ending points implies that these
properties are solely determined by the time interval. Essentially the
sequence maintains its pattern of change at any given moment.

Our exploration introduced a method that significantly improves performance.
By studying how sequence stationarity affects machine learning accuracy we
have developed a smoothing technique that not only stabilizes non stationary
sequences but also enhances the learning precision of the Long Short Term
Memory (LSTM) neural network model. This unique differential LSTM algorithm
fills gaps in current machine learning research by providing a refined
predictive framework, for time series data.

The key contribution of this research is the integration of a smoothing
technique into LSTM model training. This technique not stabilizes stationary
sequences but also boosts the learning accuracy of the LSTM neural network
model. By combining smoothing with LSTM we address existing shortcomings, in
machine learning research on time series data preprocessing offering an more
accurate predictive framework. Traditional methods, for forecasting time
series data have theories. Work well for predicting linear trends. However
they struggle when it comes to trends and large scales. In the age of data
machine learning based approaches are gaining importance for handling time
series data. Among these methods the LSTM neural network model is known for
its high prediction accuracy.

It’s important to note how improving machine learning accuracy in time series
predictions can benefit society significantly. Better forecasting can be
applied in areas like optimizing production processes and aiding policymakers
in making decisions. This research aims to contribute to enhancing machine
learning accuracy for applications, across different societal sectors.

Henceforth, to scrutinize the ramifications of sequence stationarity on
machine learning precision and enhance the prognostic acuity of machine
learning prediction methodologies, this inquiry delves into the influence of
varied non-stationary sequences on LSTM learning accuracy both pre- and post-
stabilization. The empirical validation of these postulations employs time
series data derived from the closing prices of financial market indices. The
methodological framework is as follows:

(1) Python software is employed to simulate diverse non-stationary time
series;

(2) The Time Series split cross-validation methodology partitions the
simulated non-stationary time series, facilitating the training of the LSTM
model. Subsequently, the average learning accuracy of the LSTM on the
unaltered non-stationary time series is computed;

(3) Differential smoothing and white noise testing are applied to the
simulated non-stationary time series;

(4) Utilizing the time series split cross-validation technique, the original
data and the differenced data are segregated into numerous sets for LSTM model
training and prediction. Post-de-stabilization, the average learning accuracy
of the LSTM on the time series following differential stabilization is
computed;

(5) Evaluation metrics, specifically MSE, MAE, and MAPE, are employed for a
comprehensive juxtaposition of LSTM learning accuracy on various non-
stationary time series before and after differencing. Conclusive inferences
are drawn from these analyses.;

(6) (6) To substantiate the derived conclusions, the daily closing prices of
the Shanghai Composite Index and the Shenzhen Composite Component Index are
systematically acquired through web scraping from December 31, 2012, to
December 31, 2022.

##  II Neural network related theories

###  II-A Recurrent Neural Network

Diverging from conventional artificial neural networks where inputs and
outputs operate independently, the Recurrent Neural Network (RNN) exhibits a
distinctive characteristic. Within the hidden layer of the RNN, connections
establish a temporal relationship, linking the output at any given moment to
the input from the preceding moment, thereby imbuing the network with a
discernible short-term memory. Consequently, RNN manifests as particularly
adept in processing sequentially ordered data, exemplified by its
applicability to time series data.

####  II-A1  Network structure of RNN

Given an input sequence $\\{X_{t},t=1,2,\ldots,N\\}$ and a corresponding
output sequence $\\{Y_{t},t=1,2,\ldots,N\\}$, the learning process within the
RNN unfolds through the network configuration illustrated in Figure 1. Here,
the input layer $X$ signifies the vector comprising the input sequence, $h$
denotes the hidden layer value, and $Y$ represents the vector comprising the
output sequence. The matrices, $U$, $W$, and $V$ delineate the weight matrices
governing the transitions from the input layer to the hidden layer, the
structure of the hidden layer itself, and the connectivity from the hidden
layer to the output layer, respectively.

Figure 1: Illustration of Recurrent Neural Network (RNN) Architecture  Figure
2: Diagrammatic Elaboration of Recurrent Neural Network (RNN) Structure
Expansion

Expanding the temporal network facilitates a more explicit representation of
the memory capabilities inherent in RNNs. As illustrated in Figure 2, the
output of the hidden layer at time $t$, denoted as $h_{t}$, is intricately
linked to the current input $X_{t}$ and the output $h_{t-1}$ from the
preceding time. Mathematically expressed as $h_{t}=f(U\cdot X_{t}+W\cdot
h_{t-1})$, where $f(\cdot)$ signifies the activation function, and
$Y_{t}=g(V\cdot h_{t})$, with $g(\cdot)$ representing the activation function.
Consequently, RNN establishes a continuum between input and output across
consecutive temporal instances through the interconnection of hidden layers.

####  II-A2 Advantages and Disadvantages of RNN

In contrast to conventional neural networks, recurrent neural networks exhibit
superior attributes in parameter determination and suitability for time series
data. Primarily, traditional neural network models necessitate the
determination of connection weights for each neuron, leading to
computationally intensive calculations, particularly when dealing with an
abundance of hidden layers or input layer units. In RNN, a set of weight
matrices is shared among hidden layers, substantially mitigating computational
complexity and augmenting the model’s training speed. Moreover, owing to the
cyclic interconnections within RNN’s hidden layers, it inherently possesses a
short-term memory, wherein data proximal in time exerts a more pronounced
influence on future data output. Consequently, RNN emerges as a neural network
model well-suited for the nuanced intricacies of time series data.

However, practical applications of RNN often grapple with challenges in
parameter training. During the computation of the gradient of the loss
function concerning weights, an extended time step may give rise to gradient
explosion or gradient disappearance issues. This phenomenon destabilizes the
network structure or results in the cessation of weight updates, rendering the
learning of information from a more extensive historical context unattainable.

###  II-B LSTM neutral network

In contrast to conventional RNN, the LSTM neural network structure exhibits a
distinctive capability to maintain prolonged memory and execute selective
memory operations. An extension of the original RNN architecture, the LSTM
introduces novel unit states within the hidden layer to facilitate the storage
of long-term memory, coupled with the incorporation of a gating mechanism
designed for memory selection. This unique structural augmentation effectively
addresses challenges related to the gradient explosion and vanishing gradient
issues inherent in RNN.

####  II-B1 Network structure of LSTM

As depicted in Figures 3 and 4, the LSTM network introduces a dynamic cell
state $C_{t}$ evolving over time, augmenting the original two inputs
$(X_{t},h_{t-1})$ to the hidden layer at any given time $t$ to three inputs
$(X_{t},h_{t-1},C_{t-1})$. Similarly, the outputs of the hidden layer,
originally comprising two elements $(h_{t-1},Y_{t})$, are transformed into
three outputs $(C_{t-1},h_{t-1},Y_{t})$.

Figure 3: Comparison diagram of RNN and LSTM structures Figure 4: LSTM input
and output diagram

Moreover, as delineated in Figure 5, the LSTM architecture incorporates a
forget gate, an input gate, and an output gate. These three pivotal ”switches”
play a crucial role in governing the erasure, storage, and retrieval of long-
term memory within the state $C$, respectively. Within the forget gate
mechanism, the memory state $h_{t-1}$ from the preceding time step and the
current moment’s input $X_{t}$ undergo a transformative process denoted as
$F_{t}=\sigma(W_{f}\cdot[h_{t-1},X_{t}]+b_{f})$ where $\sigma(\cdot)$
represents the sigmoid function, $W_{f}$ signifies the weight matrix
associated with the forgetting gate, $b_{f}$ represents the bias vector of the
forgetting gate, yielding a value within the range of 0 to 1 serving as the
forgetting weight. A value of 1 signifies the retention of all information,
while a value of 0 signifies complete memory erasure.

Figure 5: Schematic diagram of LSTM network structure

The input gate serves as the orchestrator for the information that undergoes
updating within the cell. Its determination is contingent upon the function
$I_{t}=\sigma(W_{i}\cdot[h_{t-1},X_{t}]+b_{i})$, where $\sigma(\cdot)$
represents the sigmoid function, $W_{i}$ denotes the weight matrix associated
with the input gate, and $b_{i}$ is the bias vector of the input gate. This
formulation entails the initiation of the input weight for the gate, enabling
the discernment of which information to update. Subsequently, a novel
candidate memory unit is conceived through the function
$\widetilde{C_{t}}=\tanh(W_{c}\cdot[h_{t-1},X_{t}]+b_{c})$. Ultimately, the
extant memory unit undergoes modification to $C_{t}=f_{t}\cdot
C_{t-1}+I_{t}\cdot\widetilde{C_{t}}$, thereby encapsulating both the
information elected for retention by the forgetting gate and the information
selected for updating by the input gate.

Concomitantly, the output gate assumes authority over the information
dissemination from the cell. Its determination adheres to the function
$O_{t}=\sigma(W_{o}\cdot[h_{t-1},X_{t}]+b_{o})$, featuring the sigmoid
function $\sigma(\cdot)$, the weight matrix $W_{i}$ for the output gate, and
the bias vector $b_{o}$. The output weight of the output gate selects the
pertinent information for output. The ultimate determinant of the output is
then disseminated through the function $h_{t}=O_{t}\cdot\tanh(C_{t})$.

The LSTM network structure, as delineated above, achieves the nuanced
management of memory through selective storage and updating mechanisms.
Additionally, it mitigates challenges such as gradient explosion and vanishing
gradients that may arise during the training process. This renders LSTM
particularly well-suited for the modeling and prediction of time series data
encountered in practical applications.

####  II-B2 A measure of learning accuracy

For the observed sequence $\\{x_{t}\\}$, let $\\{\hat{x}_{t}\\}$ denote the
predicted values, and define the error $e_{i}=x_{i}-\hat{x}_{i}$
($t=1,\ldots,n$). The subsequent error measures are frequently employed for
assessment:

Mean square error: $\text{MSE}=n^{-1}\sum_{t=1}^{n}(e_{t})^{2}$

Root mean square error: $\text{RMSE}=\sqrt{n^{-1}\sum_{t=1}^{n}(e_{t})^{2}}$

Mean absolute error: $\text{MAE}=n^{-1}\sum_{t=1}^{n}|e_{t}|$

Mean absolute percentage error:
$\text{MAPE}=n^{-1}\sum_{t=1}^{n}100\left|\frac{e_{t}}{y_{t}}\right|$

##  III Model training and impact exploration

The exploration of the influence of sequence stationarity on the accuracy of
LSTM learning is comprehensively detailed in Figure 6, encompassing distinct
stages such as the data preparation phase, the model training and prediction
phase, and the subsequent application of sequence difference smoothing and
post-prediction inverse difference operations.

Figure 6: Schematic diagram of the exploration process

###  III-A Data preparation

In the data preparation phase, Python 3.7 was utilized to simulate diverse
non-stationary sequences for investigating alterations in the learning
accuracy of LSTM networks resulting from the differentiation of distinct non-
stationary sequence types. A total of eight non-stationary sequences, each
with a length of 300 and characterized by diverse trends, were meticulously
simulated (refer to Table 1 for a comprehensive delineation of sequence types
and their corresponding expressions).

TABLE I: Simulation sequence list Sequence Type |  Sequence Expression
---|---
Random walk + stationary AR(1) sequence |  $Y_{t}=1.5Y_{(t-1)}+2\omega_{t}$
Random walk + MA(2) sequence |  $Y_{t}=Y_{(t-1)}+2\omega_{t}+0.4\omega_{(t-1)}+0.2\omega_{(t-2)}$
Random walk + stationary ARMA (1, 1) sequence |  $Y_{t}=0.6Y_{(t-1)}+2\omega_{t}+0.2\omega_{(t-1)}$
Random walk + stationary SARMA (1, 10, 1) sequence |  $Y_{t}=Y_{(t-1)}+0.2Y_{(t-10)}+2\omega_{t}+0.3\omega_{(t-10)}$
Linear trend stationary series |  $Y_{t}=0.5+0.4t+\omega_{t}$
Linear trend + stationary AR(2) sequence |  $Y_{t}=0.8+0.5t+Y_{(t-1)}-0.5Y_{(t-2)}+\omega_{t}$
Quadratic trend stationary series |  $Y_{t}=0.6+0.5t+0.2t^{2}+\omega_{t}$
Quadratic trend + stationary ARMA (1, 1) sequence |  $Y_{t}=0.6+0.8t+0.25t^{2}+0.5Y_{(t-1)}+\omega_{t}+3\omega_{(t-1)}$

The novelty in our approach becomes apparent with the introduction of
differential smoothing as part of the data preparation process. This
technique, as explained later, significantly enhances the learning accuracy of
the LSTM model, providing a performance advantage over traditional methods.
Following confirmation of non-stationarity through ADF and KPSS tests, the
sequences underwent transformation into stationary sequences via first-order
differentiation (second-order differentiation for quadratic trend sequences).
Subsequently, the LB test was executed, and the resulting p-values for each
category of sequence, both pre- and post-differentiation, are delineated in
Table 2. The outcomes ascertain that subsequent to differentiation, all
sequences exhibit stationarity and do not conform to a purely random pattern,
rendering them suitable for LSTM model training.

TABLE II: Simulated sequence stationarity and white noise test results (p value) Sequence Type | ADF (inspection) | KPSS |  LB
---|---|---|---
|  $x_{t}$ |  $\Delta x_{t}$ |  $x_{t}$ |  $\Delta x_{t}$ |  $\Delta x_{t}$
Random walk + stationary AR(1) sequence |  0.30 |  0.00 |  0.01 |  0.10 |  0.00
Random walk + MA(2) sequence |  0.87 |  0.00 |  0.01 |  0.10 |  0.00
Random walk + MA(2) sequence |  0.95 |  0.00 |  0.01 |  0.10 |  0.00
Random + stationary SARMA (1, 10, 1) sequence |  0.27 |  0.00 |  0.01 |  0.10 |  0.00
Linear trend stationary series |  0.97 |  0.00 |  0.01 |  0.10 |  0.00
Linear trend + stationary AR(2) sequence |  0.97 |  0.00 |  0.01 |  0.10 |  0.00
Quadratic trend stationary series |  0.89 |  0.00 |  0.01 |  0.10 |  0.00
Quadratic trend + stationary ARMA (1, 1) sequence |  0.97 |  0.00 |  0.01 |  0.10 |  0.00

###  III-B LSTM model training

(1) Data set division: Given the stochastic nature of the algorithm, the Time
Series Split cross-validation method is employed to partition the dataset,
including 8 non-stationary sequences and their difference-stabilized
counterparts. This strategy enables the comparison of average outcomes across
multiple training models, with specific lengths of the test set and training
set detailed in Table 3.

TABLE III: Division of simulated sequence data sets Group |  Training Set Length |  Test Set Length
---|---|---
First group |  76 |  73
Second Group |  149 |  73
The third group |  222 |  73

(2)Data conversion: Following the normalization process, the data undergoes
conversion into supervised learning data. Specifically, data at time t1 and
time t serve as input fields, while data at time t+1 functions as the output
field, facilitating the construction of samples. The array is then reshaped
into the three-dimensional input format (input length, time step, data
dimension) compatible with the LSTM model. Here, the input length corresponds
to the length of the training set and test set employed for each round of
model training. The time step is set to 2, signifying the input of data from
the initial two steps and the output of the subsequent step, with the data
dimension constrained to be one-dimensional.

(3) Model training: The establishment and training of the LSTM network are
executed within the deep learning framework tensorflow.keras. For all
experimental sequences, the number of hidden layer neuron nodes is
standardized at 12. Sequences are processed one sample at a time
($\text{batch\\_size}=1$), employing the adam optimization algorithm
($\text{optimizer}=^{\prime}adam^{\prime}$). A rigorous training regimen of
100 iterations ($\text{epochs}=100$) is implemented, culminating in the
visualization of the loss decline curve. The innovation in our approach lies
in the systematic training of the LSTM model on differentially stabilized
sequences, resulting in a considerable improvement in predictive performance
compared to traditional training methods, showcasing a clear performance
advantage.

###  III-C Comparative analysis of prediction outcomes pre- and post-
stabilization

(1) Model prediction and result visualization

Utilizing the trained LSTM models for all sequences, predictions were
generated and systematically compared with the original sequence data. The
primary objective was to visually assess the learning efficacy of the LSTM
model both before and after the application of differencing. Due to spatial
constraints, only select sequences’ prediction outcomes were showcased in the
analysis. Random Walk + Stationary AR (1) Sequence Differencing (Figure 7):
The left panel illustrates the learning outcome pre-differencing, while the
right panel illustrates the learning outcome post-preprocessing with
difference smoothing. The differential smoothing imparts stability, and post-
transformation, the LSTM model retains its capacity to capture the sequence
trend, demonstrating an improved fitting effect compared to direct utilization
of the original data.

Figure 7: Comparison of predictive outcomes pre and post the introduction of a
random walk + stationary AR(1) sequence difference.

Random Walk + Stationary ARMA (1,1) Sequence Differencing (Figure 8): The left
panel depicts the learning outcomes pre-differential processing, and the right
panel portrays the outcomes after differential stabilization preprocessing.
Evidently, subsequent to stabilization, the LSTM model exhibits superior
capacity to discern and learn the inherent trends within the sequence,
showcasing a notable advantage in learning accuracy.

Figure 8: Comparison of prediction results pre and post random walk +
stationary ARMA (1,1) sequence difference

Linear Trend Stationary Sequence Differentiation (Figure 9): The left panel
illustrates the learning outcomes before differentiation, while the right
panel portrays the learning results after preprocessing for difference
stabilization. Following the stabilization of the differentiation process, the
LSTM model adeptly assimilates inherent sequence trends, exhibiting a more
pronounced fitting efficacy compared to direct utilization of unprocessed raw
data.

Figure 9: Comparison of prediction results before and after linear trend
stationary sequence difference

Linear Trend + Stationary AR (2) Sequence Differencing (Figure 10): The left
panel shows the predictive outcome prior to differencing, juxtaposed with the
right panel showcasing the result after preprocessing for difference
smoothing. Visually discernible, the variance is mitigated, and the LSTM model
exhibits proficiency in assimilating the sequence trend even after
differencing, yielding a more robust fitting effect compared to direct
utilization of the unprocessed original data.

Figure 10: Comparison of prediction results pre and post linear trend
stationary sequence difference

The comparative outcomes for alternative sequences consistently demonstrate
that LSTM manifests superior fitting efficacy when applied to differentially
stabilized sequences. This observation leads to the plausible inference that,
in contrast to directly modeling non-stationary sequences, the preprocessing
of data through differential stabilization serves to enhance the learning
accuracy of the LSTM network.

(2) Calculate the average learning accuracy

To quantitatively assess the impact of differential stabilization on learning
outcomes, the study computes the mean learning accuracy and alteration
percentage of LSTM on the test set for each category of non-stationary
sequences, as detailed in Table 4. The tabulated data clearly indicates a
notable elevation in prediction accuracy in LSTM training subsequent to the
application of differential stabilization techniques.

TABLE IV: Average learning accuracy and change rate before and after sequence difference stabilization Serial No. | MSE | RMSE | MAE | MAPE
---|---|---|---|---
| Value | Rate of change | Value | Rate of change | Value | Rate of change | Value | Rate of change
1 | 2.48 | -29% | 1.57 | -16% | 1.30 | -19% | 0.78 | -20%
| 1.76 |  | 1.33 |  | 1.05 |  | 0.62 |
2 | 6.27 | -55% | 2.37 | -30% | 2.00 | -34% | 0.42 | 11%
| 2.80 |  | 1.67 |  | 1.31 |  | 0.47 |
3 | 5.20 | -25% | 2.27 | -14% | 1.83 | -12% | 0.96 | 5%
| 3.88 |  | 1.96 |  | 1.60 |  | 1.01 |
4 | 4.11 | -26% | 2.02 | -14% | 1.64 | -14% | 0.06 | -12%
| 3.05 |  | 1.74 |  | 1.40 |  | 0.06 |
5 | 9.50 | -85% | 3.06 | -62% | 2.73 | -65% | 0.04 | -64%
| 1.40 |  | 1.18 |  | 0.94 |  | 0.01 |
6 | 6.66 | -82% | 2.48 | -56% | 2.18 | -59% | 0.02 | -54%
| 1.18 |  | 1.09 |  | 0.89 |  | 0.01 |
7 | 21399.73 | -100% | 138.76 | -99% | 120.58 | -99% | 0.02 | -98%
| 2.59 |  | 1.59 |  | 1.32 |  | 0.00 |
8 | 145943.57 | -100% | 292.56 | -99% | 247.37 | -99% | 0.02 | -97%
| 16.05 |  | 3.97 |  | 3.22 |  | 0.00 |

Furthermore, the preprocessing involving differential stabilization manifests
a marked enhancement in the predictive performance of LSTM models,
particularly in the context of non-stationary sequences characterized by
deterministic trends such as linear and quadratic trends. This discernible
improvement underscores the efficacy of employing differential stabilization
methodologies for augmenting the accuracy of LSTM predictions in the context
of non-stationary sequences.

##  IV Example verification

###  IV-A Data sources and preprocessing

To assess the generalizability of our findings, this study acquired daily
closing price data for the Shanghai Composite Index and the Shenzhen Composite
Component Index spanning from December 31, 2012, to December 31, 2022, for the
purpose of example verification. The overall sequence length encompasses 2430
data points. Initially, both sequences underwent ADF and KPSS stationarity
tests to confirm their non-stationary nature. Subsequently, employing first-
order differencing rendered the sequences stationary, and the LB test was then
applied to ascertain their pure randomness. The p-values obtained from the
tests conducted before and after differencing are detailed in Table 5. The
outcomes affirm that the original sequences exhibit non-stationarity, which is
rectified after the initial differencing process. The utilization of white
noise sequences, indicative of non-purely random characteristics, is deemed
appropriate for training the LSTM model.

TABLE V: Stability and white noise test results of market index data (p value) Sequence | ADF(inspection) | KPSS |  LB
---|---|---|---
|  Before differential |  After differential |  Before differential |  After differential |  After differential
Shanghai Composite Index daily closing price |  0.17 |  0.00 |  0.01 |  0.10 |  0.00
Shenzhen Stock Exchange Index closing price |  0.19 |  0.00 |  0.01 |  0.10 |  0.00

###  IV-B Model training and result verification

Initially, the Time Series Split cross-validation methodology is employed to
partition the dataset into the sequence and its difference-stabilized
counterpart. This division serves as the foundation for assessing the
collective outcomes of diverse training models. The specific configurations of
the test set and training set durations are meticulously delineated in Table
6.

TABLE VI: Division of market index data set Group | Training Set Length | Test Set Length
---|---|---
First group | 609 | 607
Second Group | 1216 | 607
The third group | 1823 | 607

The training procedure of the LSTM model conforms seamlessly to the
methodology outlined in Chapter 3 and is not reiterated herein. Examination of
Figures 11 and 12 reveals that, across all test datasets, the outcomes of the
training process are geared towards enhancing the smoothness of the daily
closing price data pertaining to the Shanghai Composite Index and the Shenzhen
Composite Component Index. The predictive representation, derived post-
preprocessing and subsequently employed for LSTM model training, exhibits an
elevated degree of concordance with the authentic dataset.

Figure 11: Comparison of prediction results pre and post the daily closing
price difference of the Shanghai Composite Index Figure 12: Comparison of
prediction results pre and post the daily closing price difference of the
Shenzhen Component Index

Determine the mean learning accuracy and alteration rate of LSTM on the test
dataset both pre- and post-stabilization of the two-sequence disparity. The
outcomes are elucidated in Table 7.

sequence | MSE | rate of change | RMSE | rate of change | MAE | rate of change | MAPE | rate of change
---|---|---|---|---|---|---|---|---
The Shanghai Composite Index | 2458.82 | -33.1% | 49.33 | -18.6% | 36.73 | -26.4% | 0.01 | -26.6%
1644.30 | 40.14 | 27.03 | 0.01
The Shenzhen Composite Index | 46652.17 | -30.4% | 210.56 | -14.9% | 158.23 | -19.6% | 0.01 | -17.4%
32465.38 | 179.09 | 127.17 | 0.01
TABLE VII: Your Table Caption

The data reveals that subsequent to the stabilization of the daily closing
price difference of the Shanghai Composite Index, the prediction accuracy
employed for LSTM training experiences a reduction of approximately 33

The real-world application of our approach is exemplified through the
verification process using daily closing price data for the Shanghai Composite
Index and the Shenzhen Composite Component Index. The discernible reduction in
prediction accuracy post-differential stabilization provides real-world
evidence of the efficacy of our methodology. This practical verification
solidifies the novelty of our approach in enhancing LSTM learning accuracy for
diverse, non-stationary time series.

##  V Summary

In conclusion, this investigation into the effect of sequence smoothness on
machine learning accuracy, specifically focusing on the application of the
differential Long Short-Term Memory (LSTM) algorithm, yields significant
findings with broad implications. The study demonstrates that LSTM training
achieves heightened prediction accuracy after the stabilization of non-
stationary sequences through differential techniques. The proposed
differential-LSTM algorithm emerges as an effective means of optimizing the
original LSTM model, addressing gaps in machine learning research related to
time series data preprocessing.

Beyond the academic realm, the practical applications of these findings are
substantial. By enhancing machine learning models, especially through the
innovative differential-LSTM algorithm, the research contributes to real-world
scenarios. The improved accuracy has the potential to revolutionize
industries, support evidence-based decision-making in governance, and
ultimately benefit society at large.

While the study focuses on univariate time series stationarity, it
acknowledges the absence of exploration into the impact of multivariate time
series stationarity on machine learning accuracy. Future research directions
are proposed, encouraging the expansion of variables to enable a more
comprehensive understanding of the impact on machine learning accuracy.
Exploring interactions among different variables in a multivariate context
could offer valuable insights for practical applications where multiple
factors contribute to time series dynamics.

The suggested avenues for future research also include tailoring the proposed
differential-LSTM algorithm for specific industry applications, such as
finance, healthcare, or environmental monitoring. This customization could
enhance its applicability and contribute to advancements in domain-specific
contexts. Additionally, the integration of advanced technologies like
explainable AI and interpretability in machine learning models is recommended
to foster trust and acceptance in decision-making processes.

In summary, this investigation pioneers a novel approach to time series
analysis by emphasizing the role of sequence smoothness in optimizing machine
learning accuracy. The application of the proposed differential-LSTM algorithm
not only addresses existing research gaps but also sets the stage for positive
societal impacts through advancements in machine learning accuracy.

## References

  * [1] Livieris, I. E., Stavroyiannis, S., Iliadis, L., et al. Smoothing and stationarity enforcement framework for deep learning time-series forecasting. Neural Computing and Applications, 2021, 33(20): 14021-14035.
  * [2] Dixit, A., Jain, S. Effect of stationarity on traditional machine learning models: Time series analysis. In 2021 Thirteenth International Conference on Contemporary Computing (IC3-2021), 2021: 303-308.
  * [3] Patel, D., Canaday, D., Girvan, M., et al. Using machine learning to predict statistical properties of non-stationary dynamical processes: System climate, regime transitions, and the effect of stochasticity. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021, 31(3).
  * [4] Salazar, J. J., Garland, L., Ochoa, J., et al. Fair train-test split in machine learning: Mitigating spatial autocorrelation for improved prediction accuracy. Journal of Petroleum Science and Engineering, 2022, 209: 109885.
  * [5] Erden, G. E., Yavuz, M., Deutsch, C. V. Combination of machine learning and kriging for spatial estimation of geological attributes. Natural Resources Research, 2022, 31(1): 191-213.
  * [6] Silva, R. P., Zarpelão, B. B., Cano, A., et al. Time series segmentation based on stationarity analysis to improve new samples prediction. Sensors, 2021, 21(21): 7333.
  * [7] Livieris, I. E., Stavroyiannis, S., Pintelas, E., et al. A novel validation framework to enhance deep learning models in time-series forecasting. Neural Computing and Applications, 2020, 32: 17149-17167.
  * [8] Kamrud, A., Borghetti, B., Schubert Kabban, C. The effects of individual differences, non-stationarity, and the importance of data partitioning decisions for training and testing of EEG cross-participant models. Sensors, 2021, 21(9): 3225.
  * [9] Wang, Y., Zhang, J., Zhu, H., et al. Memory in memory: A predictive neural network for learning higher-order non-stationarity from spatiotemporal dynamics. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019: 9154-9162.
  * [10] Jamshed, A., Mallick, B., Bharti, R. K. An Analysis of Sequential Pattern Mining Approach for Progressive Database by Deep Learning Technique. In 2022 6th International Conference on Intelligent Computing and Control Systems (ICICCS). IEEE, 2022: 1409-1415.
  * [11] Herbert, Z. C., Asghar, Z., Oroza, C. A. Long-term reservoir inflow forecasts: enhanced water supply and inflow volume accuracy using deep learning. Journal of Hydrology, 2021, 601: 126676.
  * [12] Hu, H., Xia, X., Luo, Y., et al. Development and application of an evolutionary deep learning framework of LSTM based on improved grasshopper optimization algorithm for short-term load forecasting. Journal of Building Engineering, 2022, 57: 104975.
  * [13] Huang, G., Li, X., Zhang, B., et al. PM2.5 concentration forecasting at surface monitoring sites using GRU neural network based on empirical mode decomposition. Science of the Total Environment, 2021, 768: 144516.
  * [14] Sun, W., Xu, Z. A hybrid Daily PM2.5 concentration prediction model based on secondary decomposition algorithm, mode recombination technique and deep learning. Stochastic Environmental Research and Risk Assessment, 2022: 1-20.
  * [15] Kumar, M., Gupta, D. K., Singh, S. Extreme event forecasting using machine learning models. In Advances in Communication and Computational Technology: Select Proceedings of ICACCT 2019. Springer Singapore, 2021: 1503-1514.
  * [16] Ji, C., Ma, F., Wang, J., et al. Profitability related industrial-scale batch processes monitoring via deep learning based soft sensor development. Computers & Chemical Engineering, 2023, 170: 108125.
  * [17] Salam, A., Ullah, F., Amin, F., et al. Deep learning techniques for web-based attack detection in industry 5.0: A novel approach. Technologies, 2023, 11(4): 107.
  * [18] Gao, Y., Mosalam, K. M., Chen, Y., et al. Auto-regressive integrated moving-average machine learning for damage identification of steel frames. Applied Sciences, 2021, 11(13): 6084.
  * [19] Niu, D., Sun, L., Yu, M., et al. Point and interval forecasting of ultra-short-term wind power based on a data-driven method and hybrid deep learning model. Energy, 2022, 254: 124384.


# Einstein static Universe in non-minimal kinetic coupled gravity

K. Atazadeh<EMAIL_ADDRESS>F. Darabi<EMAIL_ADDRESS>Department of Physics, Azarbaijan Shahid Madani University, Tabriz, 53714-161
Iran

###### Abstract

We study the stability of Einstein static Universe, with FLRW metric, by
considering linear homogeneous perturbations in the kinetic coupled gravity.
By taking linear homogeneous perturbations, we find that the stability of
Einstein static Universe, in the kinetic coupled gravity with quadratic scalar
field potential, for closed ($K=1$) isotropic and homogeneous FLRW Universe
depends on the coupling parameters $\kappa$ and $\varepsilon$. Specifically,
for $\kappa=L_{P}^{2}$ and $\varepsilon=1$ we find that the stability
condition imposes the inequality $a_{0}>\sqrt{3}L_{P}$ on the initial size
$a_{0}$ of the closed Einstein static Universe before the inflation. Such
inequality asserts that the initial size of the Einstein static Universe must
be greater than the Planck length $L_{P}$, in consistency with the quantum
gravity and quantum cosmology requirements. In this way, we have determined
the non-minimal coupling parameter $\kappa$ in the context of Einstein static
Universe. Such a very small parameter is favored in the inflationary models
constructed in the kinetic coupled gravity. We have also studied the stability
against the vector and tensor perturbations and discussed on the acceptable
values of the equation of state parameter.

###### pacs:

04.20.Jb, 04.50.Kd, 95.36.+x

##  I Introduction

Inflationary scenario can address most of the problems in the standard
cosmology, however, in spite of the interesting prosperities of inflationary
scenario, the existence of a big bang singularity at the beginning of Universe
is the major problem of standard cosmology. In attempt to remove the initial
singularity, several theories have been proposed to address this issue, such
as the string/M-theory, the pre-big bang theory 1  and ekpyrotic/cyclic 2 .

In the static closed Friedmann-Lemaître-Robertson-Walker model, the Einstein
static Universe is one of the exact solutions of Einstein’s equations coupled
to a perfect fluid and a cosmological constant 3 . The stability conditions of
Einstein static Universe have been widely studied in the literature indicating
that this solution is not usually stable against the homogeneous perturbations
4 . In addition, it has been shown that this solution has neutral stability
against the adiabatic scalar inhomogeneities with high enough sound speed, as
well as the small inhomogeneous vector and tensor perturbations 5 .
Nevertheless, it was shown that the Einstein static Universe is unstable
against Bianchi type-IX spatially homogeneous perturbations in the presence of
perfect fluids with $\rho+3P>0$ 6 , and for various sources of matter fields 7
. Regardless of historical importance of the Einstein static Universe, the
reiterated interest to this solution comes from the “Emergent Universe”
scenario, an inflationary cosmological model in which Einstein static Universe
plays an incisive role as a initial state.

In the context of general relativity, this model was proposed in 2004 by Ellis
et al to solve the problem of initial singularity in the standard cosmological
model 8 . Moreover, the Einstein static Universe has been discussed in several
modified gravitational theories and quantum gravity models. Actually, when we
are working with the modified cosmological equations, it is possible to find
many new static solutions, essentially different from that of classical
Einstein static solution of GR, in which the stability properties depend on
the details of the studied theory or family of theories taken into account.
Basically, due to the existence of neutral stable solutions, the fine-tuning
problem of cosmological constant is so improved. But, in fact a mechanism is
needed to finish the phase of infinite expansions and collapses, and to
operate the expanding phase of the Universe 9 . Such a mechanism has been
known as “inflation” inflation .

In the context of inflationary cosmology, the role of scalar field potential
to establish an inflation is unavoidable. In general, the slowly varying
potentials behave like a large effective cosmological constant suitable for
driving an inflation. The question that “ Is it possible to recover the
cosmological constant and the inflationary phase without considering any
effective potential” led some authors to try for constructing an effective
cosmological constant starting from extended gravity theories such as non-
minimally coupled or higher order theories capo1 . In Amend , the author
considered some types of coupling between curvature and the scalar field,
called non-minimal derivative coupling. The authors in capo2  studied this
kind of couplings and connected them with inflation. Realistic cosmological
scenario was introduced based on non-minimal kinetic coupling and it was shown
that at early Universe the domination of coupling term in the field equation
predicts a quasi-de Sitter expansion 10 . In the background of cosmological
scenarios, the non-minimal kinetic coupling gravity has been considered by
choosing zero and constant potentials, for the quintessence and the phantom
cases 11 . Also, in 11 ; 111  the authors have considered some cosmological
aspects of the non-minimal kinetic coupling gravity such as Big Bang, an
expanding Universe with no beginning, an eternally contracting Universe, a Big
Crunch, a Big Rip avoidance and a cosmological bounce in the absence of the
matter. In general, the scalar tensor theory of gravity with scalar field non-
minimally coupled to gravity reveals interesting cosmological and
astrophysical behaviors Gao , capoz .

According to the above approach to the issue of inflation, it is interesting
to study the Einstein static Universe in the inflationary Universe based on
the non-minimal kinetic coupled gravity. We show that an asymptotically
Einstein static Universe in such inflationary Universe may result due to the
terms in the field equations of the non-minimal kinetic coupled gravity. In
fact, we find that at early Universe these terms could be dominating and the
cosmological evolution could have started around an Einstein static Universe
with a size $a_{0}>\sqrt{\frac{3\kappa}{\varepsilon}}$, where $\kappa$ and
$\varepsilon$ are coupling parameters (see below). We try to remove the
initial singularity problem in the standard cosmological model by studying
Einstein static Universe and its stability in the non-minimal kinetic coupled
gravity theory. Actually, the stability of Einstein static state has been
studied in various theories: in GR with a non-constant pressure ESa , in brane
world scenarios Gergely:2001tn , in Einstein-Cartan gravity Boehmer:2003iv ,
in loop quantum cosmology Mulryne:2005ef , in $f(R)$ gravity Boehmer:2007tr ;
Goswami:2008fs ; Seahra:2009ft , in Gauss-Bonnet gravity Bohmer:2009fc , in IR
modified Hořava gravity Boehmer:2009yz , in massive gravity Parisi:2012cg ,
and induced matter Brane Gravity Induced .

This paper is organized as follows. In section II, we briefly review the
formalism of the kinetic coupled gravity theory, in particular the action and
field equations. In section III, we present the modified Friedman equations
within the kinetic coupled gravity. In section IV, we consider linear
homogeneous perturbations and study the stability of Einstein static Universe
in the kinetic coupled gravity. In section V, we study the stability against
the vector and tensor perturbations. We summarize our results in section VI.

##  II Non-minimal kinetic coupling gravity

Let us consider a gravitational theory with non-minimal derivative coupling
given by the action 22

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left\\{\frac{R}{8\pi}-\big{[}\varepsilon g_{\mu\nu}+\kappa
G_{\mu\nu}\big{]}\phi^{,\mu}\phi^{,\nu}-2V(\phi)\right\\}$
$\displaystyle+S_{m},$

where ${S}_{m}$ stands for the action of matter, $V(\phi)$ is a scalar field
potential, $G_{\mu\nu}$ is the Einstein tensor, $\varepsilon$ takes the value
+1 for the canonical field and -1 for the phantom one and $\kappa$ is the
coupling parameter with dimension of (length)2. Varying the action (II) with
respect to $g_{\mu\nu}$ and $\phi$ gives the field equations, respectively:

$\displaystyle
G_{\mu\nu}=8\pi\big{[}T_{\mu\nu}^{(m)}+T_{\mu\nu}^{(\phi)}+\kappa\Theta_{\mu\nu}\big{]},$
(2a) $\displaystyle[\varepsilon g^{\mu\nu}+\kappa
G^{\mu\nu}]\nabla_{\mu}\nabla_{\nu}\phi=V^{\prime}(\phi),$ (2b)

where $V^{\prime}(\phi)\equiv dV(\phi)/d\phi$, $T^{(m)}_{\mu\nu}$ is a stress-
energy tensor of ordinary matter, and

$\displaystyle T^{(\phi)}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\varepsilon[\nabla_{\mu}\phi\nabla_{\nu}\phi-{\textstyle\frac{1}{2}}g_{\mu\nu}(\nabla\phi)^{2}]-g_{\mu\nu}V(\phi),$
(3) $\displaystyle\Theta_{\mu\nu}$ $\displaystyle=$
$\displaystyle-{\textstyle\frac{1}{2}}\nabla_{\mu}\phi\,\nabla_{\nu}\phi\,R+2\nabla_{\alpha}\phi\,\nabla_{(\mu}\phi
R^{\alpha}_{\nu)}$
$\displaystyle+\nabla^{\alpha}\phi\,\nabla^{\beta}\phi\,R_{\mu\alpha\nu\beta}+\nabla_{\mu}\nabla^{\alpha}\phi\,\nabla_{\nu}\nabla_{\alpha}\phi$
$\displaystyle-\nabla_{\mu}\nabla_{\nu}\phi\,\square\phi-{\textstyle\frac{1}{2}}(\nabla\phi)^{2}G_{\mu\nu}$
$\displaystyle+g_{\mu\nu}\big{[}-{\textstyle\frac{1}{2}}\nabla^{\alpha}\nabla^{\beta}\phi\,\nabla_{\alpha}\nabla_{\beta}\phi+{\textstyle\frac{1}{2}}(\square\phi)^{2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ -\nabla_{\alpha}\phi\,\nabla_{\beta}\phi\,R^{\alpha\beta}\big{]}.$

By imposing the Bianchi identity $\nabla^{\mu}G_{\mu\nu}=0$ and the matter
conservation law $\nabla^{\mu}T^{(m)}_{\mu\nu}=0$, equation 2a reduces to

$\nabla^{\mu}\big{[}T^{(\phi)}_{\mu\nu}+\kappa\Theta_{\mu\nu}\big{]}=0.$ (5)

Note that by inserting equations 3 and II into 5 the differential equation 5
reduces to 2b. Simply, equation 2b is a differential consequence of equation
2a.

The authors in 111 , have established an inflation model without scalar field
potential for the kinetic coupled gravity with spatially flat ($K=0$) FLRW
metric and a cosmological constant, where the cosmological evolution of
Universe at the vacuum dominated state $p_{v}=-\rho_{v}$ is described by

$a(t)\propto\exp(H_{\kappa}t),$ (6)

and

$\dot{\phi}(t)\propto\exp(-3H_{\kappa}t),$ (7)

where

$\displaystyle H\simeq\sqrt{\frac{1}{9\kappa}},$ (8)
$\displaystyle\dot{H}\simeq 0.$ (9)

As is seen in (8), the role of coupling parameter $\kappa$ in this
inflationary behavior is important such that small value of $\kappa$ results
in a sufficiently large value of the Hubble parameter. Although the present
model is different from 111 , regarding the scalar field potential and the
curvature parameter $K$, but it is interesting to study the Einstein static
Universe and its stability in the context of kinetic coupled gravity and
investigate the possible impact of stability requirement on the coupling
parameter $\kappa$.

##  III Einstein static Universe and modified Friedmann equations

###  III.1 Effective Friedmann equations

The cosmological studies of the non-minimal kinetic coupling gravity have been
sufficiently investigated 10 ; 11 ; 111 . Especially, it was shown that the
inflation and any cosmological behaviour, explicitly depends on the non-
minimally kinetic term of a scalar field $\phi$ with the curvature. However,
it is important to notice that the non-minimally kinetic term of a scalar
field $\phi$ with the curvature describes further degrees of freedom of the
gravitational field resulting from modified gravities.

We apply the Friedmann-Lemaître-Robertson-Walker (FLRW) line element as
follows

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d^{2}\theta+\sin^{2}\theta
d^{2}\phi)\right],$ (10)

where $K=+1,0,-1$ denotes a closed, flat, and open Universe, respectively.

By including the effective energy density and pressure, the modified Friedmann
equations can be written as

$\displaystyle 3H^{2}$ $\displaystyle=$ $\displaystyle 8\pi\rho_{\rm
eff}-\frac{3K}{a^{2}}\,,$ (11) $\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-4\pi\left(\rho_{\rm eff}+p_{\rm eff}\right)+\frac{K}{a^{2}}\,,$
(12)

where $\rho_{\rm eff}$ and $p_{\rm eff}$ are given by

$\displaystyle\rho_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}\Big{[}\varepsilon-3\kappa\Big{(}3H^{2}+\frac{K}{a^{2}}\Big{)}\Big{]}+V(\phi)+\rho_{m}\,,$
(13) $\displaystyle p_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}\Big{[}\varepsilon+\kappa\Big{(}2\dot{H}+3H^{2}+\frac{K}{a^{2}}+4H\ddot{\phi}\dot{\phi}^{-1}\Big{)}\Big{]}$
$\displaystyle-V(\phi)+p_{m}\,.$

The conservation equations for the matter component and the scalar field are
given by

$\displaystyle\dot{\rho}_{m}+3H(\rho_{m}+p_{m})$ $\displaystyle=$
$\displaystyle 0\,,$ (15)
$\displaystyle\varepsilon(\ddot{\phi}+3H\dot{\phi})-3\kappa\Big{[}\Big{(}H^{2}+\frac{K}{a^{2}}\Big{)}\ddot{\phi}$
$\displaystyle+2H\dot{H}\dot{\phi}+3H^{3}\dot{\phi}+\frac{KH\dot{\phi}}{a^{2}}\Big{]}$
$\displaystyle=-$ $\displaystyle V^{\prime}(\phi)\,.$ (16)

###  III.2 Einstein static Universe

To study the Einstein static Universe we impose $a=a_{0}={\rm const}$, thus
$H=\dot{H}=0$ and also we take the matter distribution that obeys from the
linear equation of state $p_{m}=w\rho_{m}$. As a result, the effective
Friedmann equations (11) and (12) can be written as

$\displaystyle 8\pi\rho_{\rm eff}$ $\displaystyle=\frac{3K}{a_{0}^{2}}\,,$
(17) $\displaystyle 4\pi\left(\rho_{\rm eff}+p_{\rm eff}\right)$
$\displaystyle=\frac{K}{a_{0}^{2}}\,,$ (18)

respectively, implying the following condition which is imposed on the
distribution of effective matter

$\displaystyle\rho_{\rm eff}+3p_{\rm eff}=0\,.$ (19)

By assuming $\phi=\phi_{0}={\rm const}$, the effective matter condition (19)
will imply

$\displaystyle\frac{1}{2}\rho_{m}^{(0)}(1+3w)=V(\phi_{0})\,,$ (20)
$\displaystyle\frac{K}{8\pi a_{0}^{2}}=\frac{V(\phi_{0})(1+w)}{(1+3w)}\,,$
(21)

and finally for the modified Klein-Gordon equation (16) we will have

$\displaystyle V^{\prime}(\phi_{0})=0,\,\,~{}~{}~{}\Lambda=8\pi V(\phi_{0}).$
(22)

Additionally, we can obtain $a_{0}$ and $\rho_{m}$ in terms of $\phi_{0}$ and
$V(\phi_{0})$.

##  IV Stability analysis of the Einstein static Universe

The Einstein static Universe has been renewed as the asymptotic inspiration of
an emergent Universe, to remove the initial singularity problem in the
inflationary cosmology 8 . Actually, these cosmological models contain
remarkable features such as the absence of an initial singularity and
avoidance of the quantum gravity intricacy. In a series of works by the
present authors the stability analysis of the Einstein static Universe has
been studied in different models. In Boehmer:2003iv , the existence and
stability of the Einstein static Universe have been studied in the Einstein-
Cartan gravity and shown that this Universe in the presence of perfect fluid
with spin density satisfying the Weyssenhoff restriction is cyclically stable
around a center equilibrium point. In Mulryne:2005ef , the stability of
Einstein static Universe against the homogeneous scalar perturbations in the
context of braneworld scenario is investigated and the stability regions are
obtained in terms of the constant geometric linear equation of state parameter
for the case of closed, open or flat Universe. It is also found that a stable
Einstein static Universe may exist in a braneworld theory of gravity against
scalar, vector and tensor perturbations for some suitable values and ranges of
the cosmological parameters. In Boehmer:2009yz , the stability of Einstein
static Universe versus the linear scalar, vector and tensor perturbations is
investigated in the context of deformed Hořava-Lifshitz (HL) cosmology
inspired by entropic force scenario. It is shown that there is no stable
Einstein static Universe for the case of flat Universe, however, for the
closed Universe and large values of running parameter of HL gravity there is
stability with domination of the quintessence and phantom matter fields, and
for open Universe there is stability with domination of the matter fields. A
neutral stability against the vector perturbations is obtained and it is shown
that for large values of the running parameter of HL gravity, there is a
stability against the tensor perturbations. In Induced , the stability of
Einstein static Universe against the scalar, vector and tensor perturbations
in the context of induced matter brane gravity is investigated. It is found
that a stable Einstein static Universe against the scalar perturbations does
exist provided that the variation of time dependent geometrical equation of
state parameter is proportional to the minus of the variation of the scale
factor. In all these works, we were motivated to find any impact of stability
requirement of Einstein static Universe on the physical characteristics of
modified theories of gravity.

Following the above mentioned line of investigation in the context of modified
theories of gravity, in the present work, first we aim to study the stability
analysis of Einstein static Universe against the homogeneous perturbations in
the kinetic coupled theory of gravity. Our motivation is to find the possible
impact of imposing such stability requirement on the kinetic coupling features
of kinetic coupled theory of gravity.

To consider the stability of the Einstein static Universe in the present
model, we take the following homogeneous perturbations

$\displaystyle a$ $\displaystyle=a_{0}+\delta a(t)\,,$ (23)
$\displaystyle\phi$ $\displaystyle=\phi_{0}+\delta\phi(t)\,,$ (24)
$\displaystyle\rho_{m}$ $\displaystyle=\rho^{(0)}_{m}+\delta\rho_{m}(t).$ (25)

Here, we only discuss on the adiabatic perturbations $\delta
p_{m}(t)=w\,\delta\rho_{m}(t)$. By perturbing the field equations (11) and
(12), and the evolution equations for the matter (15) and the scalar field
(16), we only study perturbations up to linear order.

Let us start with equation (15). Linear order perturbations gives

$\displaystyle\dot{\delta\rho_{m}}+3(1+w)\frac{\rho^{(0)}_{m}}{a_{0}}\dot{\delta
a}=0\,,$ (26)

which can be integrated as

$\displaystyle\delta\rho_{m}=-3(1+w)\frac{\rho^{(0)}_{m}}{a_{0}}\delta a\,,$
(27)

which usefully connects the matter perturbation to the scale factor
perturbation.

Now, we have three equations with three variables $\delta a$, $\delta\phi$ and
$\delta\rho_{m}$ subject to the equation (27) resulting from conservation
equation. So, we have just two independent equations to solve, namely (12) and
(16). We linearize equations (12) and (16) using the perturbations and insert
equation (27) and the background solutions (17)-(22) in the field equations
(12) and (16). The final results have been reduced to the following second
order ordinary differential equations.

$\displaystyle\ddot{\delta a}(t)-4\pi\rho^{(0)}_{m}(1+w)(1+3w)\delta a(t)=0,$
(28)
$\displaystyle\Big{(}\varepsilon-3\kappa\frac{K}{a^{2}_{0}}\Big{)}\ddot{\delta\phi}(t)+V^{\prime\prime}(\phi_{0})\delta\phi(t)=0.$
(29)

The solution of equation (28) is given by

$\displaystyle\delta a(t)=d_{1}e^{\Omega t}+d_{2}e^{-\Omega t},$ (30)

where $d_{1}$ and $d_{2}$ are the integration constants and $\Omega$ is
defined by

$\Omega=\sqrt{4\pi\rho^{(0)}_{m}(1+w)(1+3w)}~{}~{}.$ (31)

The solution is stable within the following ranges

$w>-1/3,$ (32) $-1<w<-1/3.$ (33)

Note that the interval (33) violates the strong energy condition $\rho+3p\geq
0$. To complete our study we must consider stability of the scalar field
equation (29). Thus, we must consider stability of equation (29), for $K=0$,
$K=1$ and $K=-1$. We can rewrite equation (29) as follows

$\displaystyle\ddot{\delta\phi}(t)-\frac{V^{\prime\prime}(\phi_{0})}{3\kappa\frac{K}{a^{2}_{0}}-\varepsilon}\delta\phi(t)=0,$
(34)

which provides us with the solution

$\displaystyle\delta\phi=C_{1}e^{\lambda t}+C_{2}e^{-\lambda t},$ (35)

where $C_{1}$ and $C_{2}$ are constants of integration, and $\lambda$ is
defined by

$\lambda=\sqrt{\frac{V^{\prime\prime}(\phi_{0})}{3\kappa\frac{K}{a^{2}_{0}}-\varepsilon}}~{}.$
(36)

The stability condition of the scalar field for three following cases are
given by

  * •

Flat Universe ($K=0$)

$\frac{V^{\prime\prime}(\phi_{0})}{\varepsilon}>0,$ (37)

  * •

Closed Universe $(K=1)$

$\frac{V^{\prime\prime}(\phi_{0})}{3\kappa\frac{1}{a^{2}_{0}}-\varepsilon}<0,$
(38)

  * •

Open Universe $(K=-1)$

$\frac{V^{\prime\prime}(\phi_{0})}{3\kappa\frac{1}{a^{2}_{0}}+\varepsilon}>0.$
(39)

To become more specific, we take a typical choice for the potential $V(\phi)$
and examine the stable static solutions. For instance, consider the potential
given by

$\displaystyle V(\phi)=\frac{1}{2}m^{2}\phi^{2}\,,$ (40)

where $m$ is a positive constant scalar field mass. The potential satisfies
$V^{\prime\prime}(\phi)=m^{2}$ and is strictly positive. Thus, from equations
(37), (38) and (39) we get $\varepsilon>0$,
$a^{2}_{0}>\frac{3\kappa}{\varepsilon}$ and
$a^{2}_{0}>-\frac{3\kappa}{\varepsilon}$ for the cases $K=0$, $K=1$ and
$K=-1$, respectively. It can be seen that in the case of $K=1$, the stability
condition for the scalar field results in
$a_{0}>\sqrt{\frac{3\kappa}{\varepsilon}}$. Hence, assuming
$\kappa\ll{\varepsilon}$ accounts for an small initial size for the Einstein
static Universe before the inflationary era. In particular, if we suppose that
the non-minimal coupling is set to the Planck length, by quantum gravity and
quantum cosmology considerations, as $\kappa=L_{P}^{2}$, then we find the
inequality $a_{0}>\sqrt{3}L_{P}$ which asserts that the initial size of the
Einstein static Universe must be greater than the Planck length.

##  V Vector and Tensor Perturbations

In the cosmological background, the vector perturbations of a perfect fluid
with equation of state, $p=w\rho$, are ruled by the co-moving dimensionless
vorticity defined as $\varpi_{a}=a\varpi$. Thus, the vorticity modes obey the
following propagation equation5

$\displaystyle\dot{\varpi}_{k}+(1-3c_{s}^{2})H\varpi_{k}=0\,,$ (41)

where $c^{2}_{s}=dp/d\rho$ is the sound speed and $k$ denotes the co-moving
index (see below). This equation is valid in our consideration of Einstein
static Universe in the framework of the non-minimal kinetic coupled gravity
through the modified Friedmann equations (11) and (12). For the Einstein
static background, Eq.(41) reduces to

$\displaystyle\dot{\varpi}_{k}=0\,.$ (42)

It can be seen that from the above equation, initial vector perturbations
remain frozen and therefore we have neutral stability against vector
perturbations.

Gravitational-wave perturbations, namely tensor perturbations, of a perfect
fluid is explained by the co-moving dimensionless transverse-traceless shear
$\Sigma_{ab}=a\sigma_{ab}$, whose modes satisfy the following equation

$\displaystyle\ddot{\Sigma}_{k}+3H\dot{\Sigma}_{k}+\Big{[}\frac{k^{2}}{a^{2}}+\frac{2K^{2}}{a^{2}}-\frac{8\pi}{3}(1+3w)\rho+\frac{2}{3}\Lambda\Big{]}\Sigma_{k}=0\,,$
(43)

where use has been made of $D^{2}\rightarrow-k^{2}/a^{2}$ in which $D^{2}$ is
the covariant spatial Laplacian. For the Einstein static background in our
model, this equation becomes

$\displaystyle\ddot{\Sigma}_{k}+4\pi\rho_{0}(k^{2}+2K)(1+w)\Sigma_{k}=0\,.$
(44)

In order for the Einstein static Universe becomes stable against the tensor
perturbations, and considering $\rho_{0}>0$, we have to require

$(k^{2}+2K)(1+w)>0,$ (45)

which yields the following conditions

  * •

$w>-1$ for $K=0,1$ ,

  * •

$w>-1$ for $K=-1$ and $k^{2}>2$ ,

  * •

$w<-1$ for $K=-1$ and $k^{2}<2$ .

The first two conditions show the equation of state parameter below the
phantom divide, while the latter condition shows the equation of state
parameter above the phantom divide. Considering the stability conditions (32)
and (33), it turns out that the first two conditions for stability against the
tensor perturbations are consistent with the conditions for stability against
the homogeneous scalar perturbations, except for $\omega=-1/3$. On the other
hand, the latter condition for stability against the tensor perturbations is
inconsistent with the stability conditions (32) and (33) against the
homogeneous scalar perturbations. Therefore, in an open Universe with
$k^{2}<2$, It is not possible to have stability against both scalar and tensor
perturbations.

##  VI Summary and Discussion

Non-minimal kinetic coupled gravity is one the novel modification to the
general relativity, which includes the non-minimal coupling of kinetic term of
a scalar field $\phi$ with the curvature tensor by the coupling $\kappa$, and
the minimal coupling with the metric tensor by the coupling $\varepsilon$. It
has already been shown that such modified gravity model provides an
essentially new inflationary mechanism. In this work, motivated by the idea of
“Emergent Universe” of Ellis et al, we have assumed that the Universe might
have been started out in an asymptotically Einstein static state as a initial
state before the inflationary stage of the Universe. Then, we have studied the
stability of Einstein static Universe by using linear homogeneous
perturbations in non-minimal kinetic coupled gravity. By taking a linear
equation of state parameter for the matter distribution, the stability regions
of the Einstein static Universe are specified by the second derivatives of the
scalar potential. We have shown that the stability of Einstein static
Universe, in the non-minimal kinetic coupled gravity with quadratic scalar
field potential, for closed isotropic and homogeneous FLRW ($K=1$) Universe
depends on the coupling parameters $\kappa$ and $\varepsilon$, such that in
order to have an small initial size for the Einstein static Universe,
consistent with the quantum gravity and quantum cosmology requirements (that
the Planck length is the minimum possible length for the size of the
Universe), the best choice is $\varepsilon=1$ and $\kappa=L_{P}^{2}$. Thus,
the order of magnitude of the coupling parameter $\kappa$ has been determined
in the present study of Einstein static Universe in the framework of kinetic
coupled gravity.

From the cosmological point of view, the non-minimal kinetic coupled gravity
is used for introducing new inflation models at early stage of the Universe.
In order for this theory is considered as an alternative theory of general
relativity, the contribution of the non minimal coupling should fade away for
late times, so that both theories coincide with each other at low energy
scale. The non minimal coupling includes the dominant term $\kappa
G^{00}\sim\kappa H^{2}$. During the inflationary period we have $H\gg 1$, so
that the role of $\kappa H^{2}$ is considerable in the cosmic dynamics.
However, after the inflation, when the Hubble parameter is smaller, this term
should be almost vanishing in order to recover the general relativity. Hence,
it seems the coupling parameter plays an important role to justify this
requirement. In this paper, we have found that assuming $\kappa=L_{P}^{2}$,
the stability condition imposes the inequality $a_{0}>\sqrt{3}L_{P}$ on the
initial size $a_{0}$ of the closed Einstein static Universe before the
inflation. Such inequality asserts that the initial size of the Einstein
static Universe must be greater than the Planck length $L_{P}$, in consistency
with the quantum gravity and quantum cosmology requirements. Therefore, it
seems a very small $\kappa=L_{P}^{2}$ not only provides us with a suitably
small size Einstein static Universe, but also is favored in kinetic coupled
gravity to be considered as an alternative theory of general relativity. This
may be considered as a novel cosmological viable condition imposed on the
kinetic coupled gravity theory.

We have considered a quadratic scalar field potential. If the other forms of
$V(\phi)$ are taken, the qualitative results will not change drastically.
Actually, the quadratic potential is a typical example of those potentials
which satisfy the condition $V^{\prime\prime}(\phi)>0$ at the static point
$\phi_{0}$. So, it is expected that by choosing other forms of $V(\phi)$
satisfying this condition, the qualitative results do not change drastically.

Finally, we have studied the stability against the vector and tensor
perturbations. We have found the neutral stability against the vector
perturbations, and discussed on those acceptable values of the equation of
state parameter which yield stability against both homogeneous scalar
perturbations and tensor perturbations.

## Acknowledgments

This work has been supported financially by a grant number 217/D/10856 from
Azarbaijan Shahid Madani University.

## References

  * (1) M. Gasperini and G. Veneziano, Phys. Rep. 373 (2003) 1, hep-th/0207130;
J. E. Lidsey, D. Wands and E. J. Copeland, Phys. Rep. 337 (2000) 343, hep-
th/9909061.

  * (2) J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D 64 (2001) 123522, hep-th/0103239;
P. J. Steinhardt and N. Turok, Science 296 (2002) 1436;
P. J. Steinhardt and N. Turok, Phys. Rev. D 65 (2002) 126003, hep-th/0111098;
J. Khoury, P. J. Steinhardt and N. Turok, Phys. Rev. Lett. 92 (2004) 031302,
hep-th/0307132.

  * (3) S. W. Hawking and G. F. R. Ellis, The Large scale structure of space-time, Cambridge University Press, Cambridge, 1973.
  * (4) A. S. Eddington, Mon. Not. Roy. Astron. Soc. 90 (1930) 668.
  * (5) J. D. Barrow, G. F. R. Ellis, R. Maartens and C. G. Tsagas, Class. Quant. Grav. 20 (2003) L155.
  * (6) J. D. Barrow and C. G. Tsagas, Class. Quant. Grav. 26 (2009) 195003.
  * (7) J. D. Barrow and K. Yamamoto, Phys. Rev. D 85 (2012) 083505\.
  * (8) G. F. R. Ellis and R. Maartens, Class. Quant. Grav. 21 (2004) 223;
G. F. R. Ellis, J. Murugan and C. G. Tsagas, Class. Quant. Grav. 21 (2004)
233.

  * (9) J. E. Lidsey and D. J. Mulryne, Phys. Rev. D 73 (2006) 083508;
J. E. Lidsey, D. J. Mulryne, N. J. Nunes and R. Tavakol, Phys. Rev. D 70
(2004) 063521.

  * (10) V. Mukhanov, Physical Foundations of Cosmology, Cambrdige University Press, 2005;
S. Weinberg, Cosmology, Oxford University Press, 2008.

  * (11) S. Capozziello and R. de Ritis, Gen. Relat. Grav. 29 (1997) 1425;
S. Capozziello, R. de Ritis and A. A. Marino Gen. Relat. Grav. 30 (1998) 1247;
S. Capozziello and G. Lambiase, Gen. Relat. Grav. 31 (1999) 1005, gr-
qc/9901051.

  * (12) L. Amendola, Phys. Lett. B 301 (1993) 175.
  * (13) S. Capozziello, G. Lambiase and H. -J. Schmidt, Annalen Phys. 9 (2000) 39.
  * (14) S. V. Sushkov, Phys. Rev. D80 (2009) 103505.
  * (15) E. N. Saridakis and S. V. Sushkov, Phys. Rev. D 81 (2010) 083510 .
  * (16) F. Darabi and A. Parsyia, arxiv:1312.1322.
  * (17) C. Gao, JCAP 06 (2010) 023;
L.N. Granda and W. Cardona, JCAP 1007 (2010) 021;
L. N. Granda, Class. Quantum Grav. 28 (2011) 025006;
L. N. Granda, JCAP 1104 (2011) 016;
L.N. Granda, E. Torrente-Lujan and J. J. Fernandez-Melgarejo, Eur. Phys. J. C
71 (2011) 1704;
L.N. Granda, arXiv:1109.1371;
L.N. Granda, Mod. Phys. Lett. A 27 (2012) 1250018;
H. Mohseni Sadjadi, Phys. Rev. D 83 (2011) 107301;
A. Banijamali and B. Fazlpour, Phys. Lett. B 703 (2011) 366;
G. Gubitosi, E.V. Linder, Phys. Lett. B 703 (2011) 113.

  * (18) K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, Astrophys. Space Sci. 342 (2012) 155.
  * (19) A. Ibrahim and Y. Nutku, Gen. Rel. Grav. 7 (1976) 949; C. G. Böhmer, gr-qc/0308057;
C. G. Böhmer, Gen. Rel. Grav. 36 (2004) 1039.

  * (20) L. A. Gergely and R. Maartens, Class. Quant. Grav. 19 (2002) 213;
A. Gruppuso, E. Roessl and M. Shaposhnikov, JHEP 0408 (2004) 011;
S. S. Seahra, C. Clarkson and R. Maartens, Class. Quant. Grav. 22 (2005) L91;
C. Clarkson and S. S. Seahra, Class. Quant. Grav. 22 (2005) 3653.

  * (21) C. G. Böhmer, Class. Quant. Grav. 21 (2004) 1119;
K. Atazadeh, JCAP 06 (2014) 020, arXiv:1401.7639.

  * (22) D. J. Mulryne, R. Tavakol, J. E. Lidsey and G. F. R. Ellis, Phys. Rev. D71 (2005) 123512;
L. Parisi, M. Bruni, R. Maartens and K. Vandersloot, Class. Quant. Grav. 24
(2007) 6243;
K. Atazadeh, Y. Heydarzade and F. Darabi, Phys. Lett. B 732 (2014) 223,
arXiv:1401.7638.

  * (23) C. G. Böhmer, L. Hollenstein and F. S. N. Lobo, Phys. Rev. D 76 (2007) 084005;
C. G. Boehmer, F. S. N. Lobo and N. Tamanini, Phys. Rev. D 88 (2013)104019

  * (24) R. Goswami, N. Goheer and P. K. S. Dunsby, Phys. Rev. D 78 (2008) 044011;
N. Goheer, R. Goswami and P. K. S. Dunsby, Class. Quant. Grav. 26 (2009)
105003, arXiv:0809.5247.

  * (25) S. S. Seahra and C. G. Böhmer, Phys. Rev. D 79 (2009) 064009, arXiv:0901.0892 [gr-qc].
  * (26) C. G. Böhmer and F. S. N. Lobo, Phys. Rev. D 79 (2009) 067504.
  * (27) C. G. Boehmer and F. S. N. Lobo, Eur. Phys. J. C 70 (2010) 1111;
P. Wu and H. W. Yu, Phys. Rev. D 81 (2010) 103522;
R. Canonico and L. Parisi, Phys. Rev. D 82 (2010) 064005;
Y. Heydarzade, M. Khodadi and F. Darabi, arXiv:1502.04445v1.

  * (28) L. Parisi, N. Radicella and G. Vilasi, Phys. Rev. D 86 (2012) 024035.
  * (29) Y. Heydarzade and F. Darabi, arXiv:1501.02624v1.
  * (30) S. Sushkov, Phys. Rev. D 85 (2012) 123520.