Understanding the Periodogram

A periodogram is a fundamental tool in time series analysis used to identify dominant frequencies (cycles) in the data. It estimates the spectral density of a time series, helping to detect periodic patterns.

A time series may contain cycles at different frequencies.
The periodogram helps identify which frequencies contribute most to the variance.
Peaks in the periodogram indicate dominant periodic components.

##Simulated Example

We’ll generate a simulated time series with multiple frequency components and compute its periodogram.

Steps:

Simulate a time series with known sinusoidal components.
Compute the periodogram using the Fast Fourier Transform (FFT).
Visualize the results.

set.seed(123)
library(ggplot2)

# Simulate time series with two sine waves + noise
n <- 200
t <- seq(0, 1, length.out = n)
signal <- sin(2 * pi * 5 * t) + 0.5 * sin(2 * pi * 15 * t)  # 5Hz and 15Hz components
noise <- rnorm(n, mean = 0, sd = 0.5)
time_series <- signal + noise
dft <- data.frame(t, signal, noise, time_series)
ggplot(dft, aes(x = t, y = signal)) +
  geom_line()

ggplot(dft, aes(x = t, y = noise)) +
  geom_line()

ggplot(dft, aes(x = t, y = time_series)) +
  geom_line()

# Compute periodogram
spectrum_data <- spectrum(time_series, log = "no", plot = FALSE)

# Convert frequency and power into a dataframe
df <- data.frame(Frequency = spectrum_data$freq, Power = spectrum_data$spec)

# Plot the periodogram
ggplot(df, aes(x = Frequency, y = Power)) +
  geom_line(color = "blue") +
  theme_minimal() +
  labs(title = "Periodogram of Simulated Time Series",
       x = "Frequency (Hz)",
       y = "Spectral Power")

Why Do Peaks Appear in [0, 0.1]?

The frequencies in your signal are 5 Hz and 15 Hz.
However, the frequency axis in the periodogram ranges from 0 to 0.5 (Nyquist frequency) when using normalized units.
If you are using spectrum() in R or periodogram() in Python, the frequencies are often expressed as proportions of the Nyquist frequency.

Nyquist Frequency = (Sampling Rate) / 2

If the sampling rate is 100 Hz, then:

5 Hz appears at 5/100 = 0.05

15 Hz appears at 15/100 = 0.15

So, you are seeing the 5 Hz peak in the range [0, 0.1].