Climate of any country is one of the important factors to decide the living standard of its people. It is well known that behind the development of European countries their climate has been playing a very important role. The temperature is the main parameter of climate because other parameters like rainfall, humidity, etc. depend on it directly or indirectly [1]. It is observed that in recent centuries the temperature of earth has been slightly increased. Scientists have named this increase in temperature of earth as global warming. Due to this global warming, the climate of all countries is continuously changing, so that the glaciers are melting, sea level is increasing, deserts are expanding, etc. [2]. To overcome the problems raised by global warming is one of the greatest challenges before the governments and scientists. The earth receives the energy/heat from the Sun; this energy is reflected /scattered by the earth surface and escapes into space again. Some fraction of this energy is trapped by greenhouse gases like carbon dioxide (CO2), water vapours (H2O), methane (CH4), etc. present in the earth atmosphere, which keeps earth warm and liveable. This is called the greenhouse effect due to similar working of greenhouses of America used in organic agriculture. When these greenhouse gases are produced in excess, a large amount of energy is trapped and becomes the cause of overheating of earth [3-4]. The plants, water resources and lifestyle of people of any country depend on its temperature. Due to this global warming, all the countries are bound to review their future policies [5-6]. India is a large peninsula and second highest populated country of the world. Around 17 percent of the world population is residing here. Due to the global warming effects its several megacities like Mumbai, Kolkata, Chennai, etc. are in high risk zones.
Fourier
transforms (FT) has been an important part of the signal analysis in science
and engineering. In FT the inner product of a given signal is done by
exponential functions. The FT is a very effective and useful technique to
analyze stationary signals [7]. But it is not suitable to analyze
non-stationary and transient signals. In window Fourier transforms, a window
function is taken in place of an exponential function in the inner product by
taking some part of a non-stationary signal as stationary. This is also called
short time Fourier transforms (STFT). Due to the Heisenberg uncertainty
principle, there is a restriction in time and frequency analysis by STFT [8].
In wavelet transforms, wavelet functions are taken with the signal in place of
window function in the inner product. These wavelet functions are obtained with
the dilation and translation of a single function called mother wavelet [9]. For
any two real numbers 
and 
, a wavelet
function is defined as [10]:
The continuous wavelet transforms of a function is defined as:
By taking and with , we get discrete wavelet as follows:
where the integers representing the set of discrete dilation and discrete translation, the discrete wavelet transform is defined as:
2. Discrete Wavelet Transforms (DWT):
The discrete wavelet transform is based upon wavelet, multi-resolution analysis and decomposition process of signal. The spectral analysis of any data/signal using wavelet transforms depends upon the following basic ideas:
2.1. Haar Wavelet: Haar wavelet is discontinuous, and resembles a step function. The Haar wavelet's mother wavelet function (t) can be described as:
And its scaling function can be described as:
It shows orthogonal, biorthogonal and compact support [11]. Its wavelet and scaling functions are shown in figure 1.
Figure 1: Haar Wavelet and Scaling Function
2.2 Multi-resolution Analysis (MRA):
An MRA consists of a sequence of closed subspaces of L2(), a Lebesgue space of square integrable functions, satisfying the following properties [12-13]:
1. For every
2. There exists a function such that is orthonormal basis of .
The function is called scaling function of given MRA and property 3 implies a dilation equation as following:
where is low pass filter and is defined as:
The wavelet functionis expressed as:
where is high pass filter and is defined as:
2.3 Decomposition of signal:
From conditions of multi-resolution analysis (MRA) and elementary functional analysis, each space can be decomposed into subspace and such that every function in can be uniquely decomposed into with and . We write this,
If
all such functions 
and 
are orthogonal (), then is the
orthogonal complement of in and the construction below will give the scaling
function and mother wavelet of an orthonormal wavelet basis for [14]. By MRA, the orthogonal decomposition of space is as following:
where subspace
With help of above theory of MRA, any function or signal can be expressed by first order decomposition as follows [15]:
By second order decomposition, it can be expressed as:
In general, for order decomposition, a signal can be expressed as:
Where represents to the order of decomposition of signal or level of the wavelet transforms, where:
are collectively known as approximation and detailed coefficients of the given signal [10]. Thus, a given signal takes place a new version such as:
Here is approximation and is detail of signal at th level or time frames. Taking , i.e. putting in equation (5), a signal can be expressed as:
A signal can be decomposed as in simplest form (level 1):
Taking we can write,
Figure 2: Signal Decomposition (First Order)
Figure 3: Signal Decomposition ((J+1) Th Order)
3. Stationary Wavelet Transforms (SWT):
Discrete wavelet transforms (DWT) is not a time-invariant transform, that is why, cannot be applied for the extension of any signal. The SWT algorithm is designed to overcome the lack of translation-invariance of the (DWT). By upsampling the filter coefficients by a factor of in the th level of the algorithm, the translation invariance is achieved. For SWT, the redundant approximation and detail are obtained as follows:
where ∈
{0, … . −
1} allows for all the possible shifts in a discrete setting. For decomposition
to levels,
2 different
orthogonal bases are generated. Each node in binary tree is indexed by
parameters (𝑗, 
),
to which the set of coefficients is
associated. The SWT is identical to the standard wavelet transforms and only
differs in terms of shift and decimation. The general step 
convolves
the approximation coefficients at level 
with
upsampled versions of the appropriate original filters to obtain the
approximation and detail coefficients at level [16].
Figure 4: Upsampling of Approximation and detail coefficients
4. Research Methodology:
We have taken average temperature of India from 1901-2019 as original data (Imported from website data.gov.in). Its quantitative behaviour is shown in the figure 5.
Figure 5: Quantitative Behaviour of Average Temperature of India from 1901-2019
For spectral analysis and forecasting of the given data of temperature, we use a wavelet toolbox of MATLAB software [17]. We select the best wavelet for this signal using an adaptive wavelet process [18]. The Haar wavelet is the adaptive wavelet for the given signal. The stationary wavelet transform is performed to maintain the translation invariance of the signal. The signal is extended up to 2028 data points and the average behaviour of average temperature of India with the coming 50 years is obtained. The discrete wavelet transform of signals are performed for spectral analysis of the signal. The original and extended signals are decomposed into approximations and details. The highest scale approximation represents the average behaviour or trend of the signal. The trend of both original signal and extended signal are revealed. The statistical parameters like average, skewness and kurtosis of the original and extended signal are determined and discussed [19-20].
5. RESULTS AND DISCUSSION:
By using symmetric padding stationary wavelet transforms, the signal corresponding to average temperature record of India from 1901-2019 is extended for 50 years. The original and extended signals are shown in red and yellow blocks respectively in figure 6.
Figure 6: Original and Extended Average Temperature
The maximum possible scale resolution describes the average behaviour or trend of the signal. In figure 7, the average behaviour of the original signal at scale-9 (Trend of average temperature behaviour of India from 1901-2019) is shown.
Figure 7: Trend of original signal at level-9
In figure 8, the average behaviour of extended signal (Trend of average temperature behaviour of India from 1901-2069) on the same resolution scale (level-9) is shown.
Figure 8: Trend of extended signal at level-9
In figure 9, the average behaviour of extended signal (Trend of average temperature of India from 1901-2069) on the next higher resolution scale (level-10) is shown.
Figure 9: Trend of extended signal at level 10
It is clear from this trend of the signal that the effect of global warming in India is continuously increasing with a slight declined rate. The statistical parameters of original and extended signals are given in table1.
Table 1: Statistical parameters of original and extended signal
|
S. No. |
Statistical Parameter |
Original Signal |
Extended Signal |
|
1 |
Average |
24.32919 |
24.2542 |
|
2 |
Skewness |
-0.45176 |
-0.44243 |
|
3 |
Kurtosis |
-1.223 |
-1.22227 |
The average value of the extended signal is slightly less than that of the original signal, which indicates the lack of effect of global warming in India. Skewness measures asymmetry of the data point about mean value and its negative value indicates that the data is skewed left. The value of skewness for original and extended both signals are negative but the value for extended signal is slightly less than that of original signal, which indicates that there will be a slight decrease in the rate of global warming in the coming 50 years. The kurtosis measures the peakedness of data points about mean position. The negative and small value of kurtosis indicates that the data of average temperature of India in the coming years will be slightly flat than that of present time.
6. CONCLUSION:
The average temperature record of India from 1901-2019 is taken as raw data and extended up to 50 years (2069) using symmetric padding stationary wavelet transforms. The trend or average behaviour of the signal corresponding to the highest scale approximation shows continuous increase in the temperature of India. From the trend of the original and extended signal it is revealed that in the coming 50 years there will be a slight decrease in the rate of the average temperature of India. The statistical analysis of the average temperature of India (Original and extended) provides strong consistency with its spectral analysis using wavelet transforms. Obviously, the spectral and statistical analytical results are the strong evidence in the favour of continuous increase in effects of global warming in India with a slight declined rate. The discrete and stationary wavelet transforms provide a simple and accurate framework for modelling the quantitative behaviour of observed temperature of India and its extension.
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Received on 10.09.2020 Modified on 29.09.2020 Accepted on 12.10.2020 ©A and V Publications All right reserved Research J. Science and Tech. 2020; 12(4):260-266. DOI: 10.5958/2349-2988.2020.00035.2 |
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