Spectral Analysis in R: Periodogram and Power Spectrum

Spectral analysis rewrites a time series as a mix of repeating waves and measures how strong each one is. The periodogram is the chart that shows which cycle dominates, so you can spot a hidden 12-month season or an 11-year sunspot rhythm at a glance. Every example here is base R and runs right in your browser.

What does spectral analysis tell you about a time series?

A line chart shows a series wiggling up and down, but it rarely tells you how long one full up-and-down cycle takes, or whether two cycles are stacked on top of each other. Spectral analysis answers exactly that. Let us plant a cycle whose length we already know, bury it in random noise, and watch a single function recover it.

We will build 144 points made of one clean wave that repeats every 12 steps, plus a generous helping of noise. Then we hand the whole thing to spectrum() and ask it a simple question: which cycle length carries the most energy? The answer should come back as 12, even though the noise makes the raw series look messy.

Here is what each line did. We seeded the random generator so you get the same numbers, built a sine wave that completes one cycle every 12 steps, and added rnorm() noise so the pattern is not obvious to the eye. spectrum() computed how much power sits at each frequency and stored the result in sp, with the tested frequencies in sp$freq and their power in sp$spec. which.max(sp$spec) found the position of the strongest peak, and dividing 1 by the frequency at that position converted it into a period. The result is 12, the exact cycle we planted.

That is the whole promise of spectral analysis. You give it a noisy series, and it hands back the length of the strongest repeating cycle. The rest of this tutorial unpacks how that number is computed, how to read the full picture instead of just the top peak, and how to trust the result on real data.

What are frequency, period, and cycles in a time series?

Before we read a single chart, we need to pin down three words that spectral analysis leans on constantly. A cycle is one full repeat of a pattern, like winter-to-winter in monthly temperatures. The period is how many time steps that one repeat takes. The frequency is how many repeats happen per single time step.

Period and frequency are two views of the same thing, and they are simple reciprocals of each other.

$$T = \frac{1}{f}$$

Where:

• $T$ is the period, measured in time steps per cycle
• $f$ is the frequency, measured in cycles per time step

A slow cycle has a long period and a low frequency. A fast cycle has a short period and a high frequency. The table below makes the trade-off concrete for three frequencies.

Read the first row like this: a frequency of 0.05 means the wave gets through 0.05 of a cycle each step, so it needs 20 steps to finish one full cycle. A frequency of 0.25 is four times faster, so it wraps up a cycle in just 4 steps. The periodogram always reports frequency on its x-axis, so you will convert to a period constantly, and it is always the same one-over-frequency move.

How do you read a periodogram in R?

The periodogram is the core chart of spectral analysis. Its x-axis is frequency and its y-axis is power, which you can think of as how much of the series' wiggle each frequency is responsible for. A tall spike means a strong, dominant cycle at that frequency. A flat, low region means nothing much is happening there.

When you call spectrum(y), R draws this chart for you. It also stores the raw numbers in the returned object: sp$freq holds every frequency it tested, and sp$spec holds the power at each one. Instead of squinting at the plot, we can sort those numbers and print the five strongest frequencies as a table.

Look at the power column. The top row has a power of 237.74 at a period of 12, and every other row sits below 17. That gap is the story: one cycle towers over everything else. The runner-up frequencies are not real cycles at all, they are just the noise leaving faint ripples across the chart. This is exactly how you separate signal from noise in a periodogram: hunt for the peak that stands far above its neighbours.

How does the periodogram work under the hood (the FFT)?

So far spectrum() has been a black box. Let us open it, because the machinery is simpler than it sounds and understanding it will help you trust the output. The engine is the Fast Fourier Transform, available in base R as fft(). It takes your series and reports, for every frequency, how strongly a wave at that frequency matches your data.

The periodogram is then just the squared size of that match, scaled by the length of the series. We will do it by hand: subtract the mean so the flat "zero-frequency" part does not dominate, run fft(), square the magnitude with Mod(), divide by n, and keep only the first half of the frequencies (we will see why the second half is redundant in the next section).

Walk through it slowly. yc is the series with its average removed. fft(yc) returns a complex number per frequency, and Mod() gives the length of each of those complex numbers, which is the strength of the match. Squaring and dividing by n turns that into power, freq lists the frequencies that go with each entry, and half keeps the meaningful lower half. The strongest of those lands at frequency 1/12, so peak_period is 12, the same answer spectrum() gave us. The black box was never mysterious: it is just a Fourier transform, then a square and a scale.

Here is the same idea written as a formula. If you would rather not read math, skip to the next section, because the code above already told the whole story.

$$I(f_j) = \frac{1}{n}\left| \sum_{t=1}^{n} x_t \, e^{-2\pi i f_j t} \right|^2$$

Where:

• $I(f_j)$ is the periodogram value (the power) at frequency $f_j$
• $x_t$ is the series value at time $t$
• $n$ is the number of observations
• the sum inside the bars is the discrete Fourier transform, which fft() computes for every frequency at once

The diagram below traces that pipeline from raw series to cycle length.

Figure 1: How the FFT turns a time series into a periodogram and a cycle length.

Which cycles can you actually detect? Fourier frequencies and the Nyquist limit

A periodogram does not test every imaginable frequency. It tests a fixed grid called the Fourier frequencies, which are the fractions $j/n$ for $j$ running from 1 up to $n/2$. Two limits fall straight out of that grid, and knowing them keeps you from chasing cycles your data cannot possibly show.

$$f_j = \frac{j}{n}, \quad j = 1, 2, \ldots, \frac{n}{2}$$

The lowest frequency is $1/n$, which is one single cycle stretched across the entire record. You cannot detect anything slower, because a slower cycle would not even complete once in your data. The highest frequency is $1/2$, called the Nyquist frequency, which is the fastest cycle you can see: it takes two observations to define one up-and-down swing. The block below lists these limits for our 144-point series.

The table spells out the boundaries. With 144 points you get 72 frequencies to inspect, the longest detectable cycle is 144 steps (the whole series is one cycle), and the shortest is 2 steps. Anything faster than the Nyquist limit is invisible, and worse, it appears in the spectrum as a slower cycle, an error called aliasing.

How do you find the dominant cycle in real data?

Planted signals are reassuring, but the real test is messy data with a cycle nobody wrote in. The classic example is sunspot.year, the yearly count of sunspots stretching back to 1700. Astronomers have long known the Sun runs on an eleven-year rhythm, and spectral analysis should rediscover it without being told. We will pull the top five peaks just as before.

The top row nails it: a period of 11.1 years with far more power than anything else, exactly the solar cycle physicists talk about. Notice the neighbours at 10.0 and 10.3 years, which are the same broad peak spread across adjacent frequencies, not separate cycles. The 100-year entry in row three is a slow drift in overall activity, real but weaker. Because sunspot.year is stored as an annual time series, R reports frequency in cycles per year, so the periods come out directly in years.

Why is the raw periodogram noisy, and how does smoothing build the power spectrum?

If you plot the raw periodogram of real data, it looks like a spiky mess even around a clear peak. That is not a display glitch, it is a genuine flaw: the raw periodogram is a noisy estimator, and its noise does not shrink as you collect more data. More points give you more frequencies, but each one stays just as jittery. The fix is to smooth neighbouring frequencies together, which trades a little precision on cycle length for a much steadier, more trustworthy estimate. That smoothed curve is what people usually mean by the power spectrum or spectral density.

In R, you smooth by passing the spans argument to spec.pgram(). It averages each frequency with its neighbours using a gentle weighting (the modified Daniell kernel). Below we compare the raw estimate against a smoothed one for the sunspot series, printing each one's peak period, bandwidth and degrees of freedom.

Read the two rows against each other. Smoothing barely moved the peak, from 11.11 to 10.71 years, so the eleven-year cycle survives, which is the reassurance you want. The bandwidth grew from 0.001 to 0.006, meaning the estimate now blurs each frequency over a wider window, and the degrees of freedom jumped from about 2 to about 11, which is why the smoothed curve is so much steadier. The default spectrum() plot draws a small cross showing this bandwidth and a confidence interval, so you can judge whether a peak is tall enough to be believed.

How do trends and leakage distort the spectrum, and how do you fix them?

Real series often drift upward or downward over time, and that trend wrecks a naive periodogram. A steady climb looks, to the Fourier transform, like the first slow half of an enormous low-frequency cycle, so it puts most of the power at the lowest frequencies and buries every real cycle underneath. The famous AirPassengers series, monthly airline totals from 1949 to 1960, climbs steeply and shows the problem clearly. We compute its spectrum twice, once without removing the trend and once with detrend = TRUE.

The contrast is stark. Without detrending, the tallest peak sits at a 12-year period, which is simply the full length of the record: the trend dominates the spectrum and shows up as one giant cycle. After detrending, the peak snaps to a 1-year period, which is 12 months, the genuine annual travel season everyone expects. This is why spectrum() and spec.pgram() detrend by default, and it is the single most common mistake beginners make when they turn that default off.

The taper argument fixes a subtler problem called leakage. Because your record is finite, it usually chops a cycle off partway through, and that abrupt edge smears a strong peak's energy into nearby frequencies. Tapering gently fades the two ends of the series toward its mean before the transform, which softens the edges and sharpens the peak. The default of taper = 0.1 fades the first and last ten percent, which is a sensible starting point.

Complete Example

Let us put every step together on a fresh series and finish with something practical: finding a cycle, then filtering it out. The nottem dataset holds monthly average temperatures at Nottingham Castle from 1920 to 1939. It has an obvious yearly rhythm, so it is the perfect series to find a cycle in and then remove.

First we locate the dominant cycle exactly as before. Because nottem is monthly, we convert the period into months by multiplying by 12.

The dominant cycle is one year, or 12 months, the summer-to-summer temperature swing. Now for the useful part. Once you know a cycle's length, you can subtract it out and study whatever is left. We fit a sine and a cosine at the annual frequency with lm(), a technique called harmonic regression, and keep the residuals. Removing the annual wave should slash the variance and shift the dominant peak of what remains.

The numbers tell a clean story. Before filtering, the series had a variance of 73.48 dominated by a 12-month cycle. After subtracting the annual wave, the variance collapsed to 6.42, meaning the yearly season alone accounted for about 91 percent of all the movement. The dominant cycle of the leftovers jumped from 12 months to 72 months, a slow six-year drift that was completely hidden under the seasonal swing. That is spectral analysis earning its keep: it found the cycle, and using its frequency we filtered it away to reveal a second, subtler pattern underneath.

Practice Exercises

These combine several ideas from the tutorial. Try each one before opening the solution. The variables use a my_ prefix so they will not clash with anything above.

Exercise 1: Recover two hidden cycles at once

Build a 240-point series that hides two cycles, one with a period of 8 and one with a period of 20, plus light noise. Then recover both by reading the top two periodogram peaks. Expected output: the two periods 8 and 20.

Exercise 2: Compare a raw and a smoothed spectrum on real data

Using nottem, find the dominant period (in months) from the raw spectrum and from a smoothed spectrum with spans = c(3, 3), and report the smoothed bandwidth. Confirm the annual peak survives smoothing.

Exercise 3: Prove that fft() and spectrum() agree

Generate a cosine wave with a period of 25 plus noise, then find the dominant period two ways: by hand with fft(), and with spectrum(). Show that both return the same period.

Frequently Asked Questions

What is the difference between a periodogram and a power spectrum?

The periodogram is the raw estimate: the squared, scaled output of the Fourier transform at every frequency. It is spiky and noisy. The power spectrum (or spectral density) is the smoothed version you get after averaging neighbouring frequencies with spans. In everyday use people treat the smoothed curve as the thing worth reading, because the raw periodogram's noise never settles down no matter how much data you have.

Should I use spectrum() or spec.pgram()?

Use spectrum() for a quick look; it is a friendly wrapper that detrends and tapers the series before plotting it with sensible defaults. Reach for spec.pgram() when you want direct control over the arguments, especially spans for smoothing, taper for leakage, and detrend for trend removal. They share the same engine, so the peak you find will agree.

Why is my biggest peak stuck at the lowest frequency?

That is almost always an untreated trend. A series drifting up or down looks like the slow start of one giant cycle, so it piles power at the lowest frequencies. Detrend the series first (it is the default in spectrum()), and the real cycles will surface, exactly as they did for AirPassengers above.

How much data do I need to detect a cycle?

You need at least two full repeats of the cycle just to register it, and several more to trust the peak. A yearly cycle in monthly data (period 12) needs a bare minimum of 24 months, but two or three years give a far cleaner signal. The longer your record, the finer the frequency grid, so longer series also separate close cycles more clearly.

Can I run spectral analysis on unevenly spaced data?

Not with fft(), spectrum(), or spec.pgram(), which all assume evenly spaced observations. For irregular timestamps or missing points, the standard tool is the Lomb-Scargle periodogram, available in the lomb package, which estimates the spectrum directly from the time-value pairs without requiring a fixed sampling interval.

Summary

Spectral analysis turns a confusing time plot into a clear ranking of cycles. You compute a periodogram, read its tallest peak, and invert the peak's frequency to get the dominant cycle length. The workflow below captures the five moves you will repeat every time.

Figure 2: The five-step spectral analysis workflow, from detrending to reading the peak.

Keep the flow in Figure 2 close: detrend, compute the periodogram, smooth it, read the peak, and convert the frequency to a period. With those five steps you can uncover a hidden season, confirm a known rhythm, or filter a cycle out to see what lies beneath.

References

1. R Core Team. spec.pgram: Estimate Spectral Density of a Time Series by a Smoothed Periodogram. R stats package documentation. Link
2. R Core Team. spectrum: Spectral Density Estimation. R stats package documentation. Link
3. Shumway, R. H., & Stoffer, D. S. Time Series Analysis and Its Applications: With R Examples, 4th Edition. Springer (2017). Chapter 4: Spectral Analysis and Filtering. Link
4. Cryer, J. D., & Chan, K.-S. Time Series Analysis: With Applications in R, 2nd Edition. Springer (2008). Chapter 13: Introduction to Spectral Analysis. Link
5. PennState STAT 510. Lesson 6: The Periodogram. Applied Time Series Analysis. Link
6. R Core Team and contributors. CRAN Task View: Time Series Analysis (see the Frequency Analysis section). Link

Continue Learning

• Test for Stationarity in R: Spectral analysis assumes a stable series, so check stationarity with the ADF and KPSS tests before you trust the peaks.
• ACF and PACF in R: The time-domain cousin of the periodogram; autocorrelation finds repeating structure by comparing a series with lagged copies of itself.
• Time Series Decomposition in R: Once you know the dominant cycle, split the series into trend, seasonal, and remainder components to model each piece on its own.