Spectral Analysis of Time Series Data

The concept of regularity in time series refers to the presence of consistent, repeating patterns or structures over time within a sequence of data points. In time series analysis, regularity indicates that certain events or behaviors occur at predictable intervals, which can be associated with seasonal or cyclical phenomena. It suggests the data generating procedure has a deterministic component.

To fully describe a sine curve, three constants are required: amplitude, frequency, and phase. In the context of a sine curve, amplitude refers to the maximum displacement or distance moved by a point on the wave from its equilibrium position, the X-axis. In simpler terms, it represents the wave’s height and signifies the signal’s intensity or strength.

Frequency denotes the number of complete cycles or oscillations that occur within a given unit of time. It determines the pitch in sound waves and the color in light waves. A higher frequency corresponds to faster oscillations, whereas a lower frequency indicates slower oscillations. Frequency is inversely proportional to wavelength.

Phase describes the position of a point in time within the wave’s cycle. It represents how much the wave is horizontally shifted relative to a reference point or another wave. Phase is typically measured in degrees (0° to 360°) or radians, and it reflects the wave’s position in its cycle compared to a specified reference.

import numpy as np
import matplotlib.pyplot as plt

# Generate a sine wave
Fs = 500  # Sampling frequency
f = 5  # Frequency of the sine wave in Hz
x = np.linspace(0, 2, 2 * Fs)  # Time from 0 to 2 seconds, sampled at Fs
y = np.sin(2 * np.pi * f * x)  # Sine wave

# Plot the sine wave
plt.figure(figsize=(12, 6))

# Time domain plot (sine wave)
plt.subplot(2, 1, 1)
plt.plot(x, y)
plt.title('Sine Wave (Time Domain)')
plt.xlabel('Time [s]')
plt.ylabel('Amplitude')

# Frequency domain representation using Fourier Transform
Y = np.fft.fft(y)
freqs = np.fft.fftfreq(len(y), 1 / Fs)

# Plot the magnitude spectrum
plt.subplot(2, 1, 2)
plt.plot(freqs[:len(freqs) // 50], np.abs(Y)[:len(freqs) // 50])  # Plot only positive frequencies
plt.title('Frequency Domain Representation')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Magnitude')

plt.tight_layout()
plt.show()

Now we shall consider a mixture of two sine curves and we shall find the frequency domain representation of the same.

import numpy as np
import matplotlib.pyplot as plt

# Clear the previous plots
plt.clf()

# Generate a sine wave
Fs = 500  # Sampling frequency
amp1 = 1
f1 = 5  # Frequency of the first sine wave in Hz
x = np.linspace(0, 2, 2 * Fs)  # Time from 0 to 2 seconds, sampled at Fs
y1 = amp1 * np.sin(2 * np.pi * f1 * x)  # First sine wave

amp2 = 4
f2 = 3  # Frequency of the second sine wave in Hz
y2 = amp2 * np.sin(2 * np.pi * f2 * x)  # Second sine wave

# Superimpose the two sine waves
y = y1 + y2

# Plot the sine wave
plt.figure(figsize=(12, 6))

# Time domain plot (sine wave)
plt.subplot(2, 1, 1)
plt.plot(x, y)
plt.title('Sine Wave (Time Domain)')
plt.xlabel('Time [s]')
plt.ylabel('Amplitude')

# Frequency domain representation using Fourier Transform
Y = np.fft.fft(y)
freqs = np.fft.fftfreq(len(y), 1 / Fs)

# Plot the magnitude spectrum
plt.subplot(2, 1, 2)
plt.plot(freqs[:len(freqs) // 50], np.abs(Y)[:len(freqs) // 50])  # Plot only positive frequencies
plt.title('Frequency Domain Representation')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Magnitude')

plt.tight_layout()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

# Clear the previous plots
plt.clf()

# Generate a sine wave
Fs = 5000  # Sampling frequency
x = np.linspace(0, 2, 2 * Fs) 
amp = [1,5,3,8,12]
freq = [5,8,12,19,6] 
y=[0]*len(x)
for i in range(len(amp)):
  y1 = amp[i] * np.sin(2 * np.pi * freq[i] * x) 
  y=y+y1

# Plot the sine wave
plt.figure(figsize=(12, 6))

# Time domain plot (sine wave)
plt.subplot(2, 1, 1)
plt.plot(x, y)
plt.title('Sine Wave (Time Domain)')
plt.xlabel('Time [s]')
plt.ylabel('Amplitude')

# Frequency domain representation using Fourier Transform
Y = np.fft.fft(y)
freqs = np.fft.fftfreq(len(y), 1 / Fs)

# Plot the magnitude spectrum
plt.subplot(2, 1, 2)
plt.plot(freqs[:len(freqs) // 250], np.abs(Y)[:len(freqs) // 250])  # Plot only positive frequencies
plt.title('Frequency Domain Representation')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Magnitude')

plt.tight_layout()
plt.show()

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stat

# Clear the previous plots
plt.clf()

# Generate a sine wave
Fs = 5000  # Sampling frequency
x = np.linspace(0, 2, 2 * Fs) 
amp = [1,5,3,8,12]
freq = [5,8,12,19,6] 
y=[0]*len(x)
for i in range(len(amp)):
  y1 = amp[i] * np.sin(2 * np.pi * freq[i] * x) 
  y=y+y1
y=y+stat.norm.rvs(scale=10,size=len(y))
# Plot the sine wave
plt.figure(figsize=(12, 6))

# Time domain plot (sine wave)
plt.subplot(2, 1, 1)
plt.plot(x, y)
plt.title('Sine Wave (Time Domain)')
plt.xlabel('Time [s]')
plt.ylabel('Amplitude')

# Frequency domain representation using Fourier Transform
Y = np.fft.fft(y)
freqs = np.fft.fftfreq(len(y), 1 / Fs)

# Plot the magnitude spectrum
plt.subplot(2, 1, 2)
plt.plot(freqs[:len(freqs) // 250], np.abs(Y)[:len(freqs) // 250])  # Plot only positive frequencies
plt.title('Frequency Domain Representation')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Magnitude')

plt.tight_layout()
plt.show()

Primary objective of spectral analysis is decompose a time varying quantity into a sum of sine and cosine functions.

Sine and cosine functions are periodic with periodicity of \(2\pi\). Then any linear combination of these functions is also periodic with the same period \(2\pi\). Consider the function \[X_t=\sum_{n=1}^{\infty} \left(\alpha_n sin nt+\beta_n cos nt\right)\] where \(\left\{\alpha_k\right\}, \left\{\beta_k\right\}\)