Zaman Serisi Analizine Giriş: Bileşenler, Durağanlık ve ARIMA

The moment a time stamp is attached to a column of numbers, the problem changes. Observations are no longer independent: today's value depends on yesterday's, and this month's on the same month last year. The “observations are independent” assumption behind classical regression breaks down, and the data calls for its own toolkit.

In this post I work through the core ideas of time series analysis in R: decomposing a series into its components, stationarity, autocorrelation, and forecasting with ARIMA. All examples use nottem, a data set that ships with R, so you can reproduce them without downloading anything.

“The real question in time series analysis is not ‘which model is better?' but ‘which part of this series is systematic and which part is chance?'”

1. What Is a Time Series?

A time series is data formed by ordering observations of a variable by time. With T as the sample size, it is usually written as X_t, t = 1, 2, …, T.

In R, what turns a vector into a time series is the ts class and its frequency. Frequency says how many observations make up one seasonal cycle: 12 for monthly data, 4 for quarterly. Every seasonal calculation that follows rests on this single number.

library(tidyverse)
library(forecast)
library(tseries)

# nottem: monthly average air temperature at Nottingham Castle (°F), 1920-1939
class(nottem)      # "ts"
frequency(nottem)  # 12 -> monthly
start(nottem)      # 1920 1

autoplot(nottem) +
  labs(title = "Monthly average temperature in Nottingham",
       x = "Year", y = "Temperature (°F)")

2. The Components of a Series

A time series combines deterministic elements, which can be written as a formula, with stochastic elements that arise from random processes. Classical decomposition reduces this structure to four components.

These components can combine in two ways. The additive model fits when the amplitude of the seasonal swing is independent of the level of the series:

The multiplicative model is used when the seasonal swing grows along with the level:

3. Decomposition with decompose()

In R, decompose() estimates the trend with moving averages, extracts seasonal averages from the detrended series, and returns what is left as the random component.

parts <- decompose(nottem)        # default: type = "additive"
plot(parts)

# Accessing the components one by one
parts$trend
parts$seasonal
parts$random

# Multiplicative decomposition
plot(decompose(nottem, type = "multiplicative"))

The chart has four panels: the observed series, the trend, the seasonal component and the random component. For nottem, the seasonal panel shows a wave that repeats almost exactly, while the trend panel has no clear direction — across 20 years the average temperature moves within a narrow band.

Because the amplitude of the seasonal wave does not grow over the years, the additive model is the right choice for this series. The classic example of multiplicative behaviour is AirPassengers: as passenger numbers rise, the gap between summer and winter grows in absolute terms, so the amplitude depends on the level.

4. Stationarity

In a stationary series the mean, the variance and the covariance at given lags do not change over time. Whatever window you cut the series at, its statistical behaviour is the same.

Why does this matter? Because forecasting rests on the assumption that behaviour observed in the past will continue to hold. In a non-stationary series that assumption fails: the model fits the past but does not carry into the future. On top of that, two non-stationary series can produce a high R² with no real relationship between them at all — a spurious regression.

The random walk is the best-known example of a non-stationary series:

Here et is white noise. The model itself is not stationary, because every shock is added permanently to the level of the series. The most common way to make a series stationary is differencing: taking the difference between consecutive observations reduces a random walk to white noise.

5. Autocorrelation: ACF and PACF

While correlation measures the linear relationship between two variables, autocorrelation measures the relationship of a series with its own lagged values. rk is the relationship between the series and its k-lagged version:

Plotting these coefficients against the lag gives the correlogram (ACF). The partial autocorrelation (PACF) shows the direct relationship that remains once the effect of the intermediate lags has been removed.

ggAcf(nottem, lag.max = 25) +
  labs(title = "nottem — autocorrelation (ACF)")

ggPacf(nottem, lag.max = 42) +
  labs(title = "nottem — partial autocorrelation (PACF)")

The blue dashed lines are the confidence bounds. When reading these plots, look for:

6. Unit Roots and the ADF Test

Plots and correlograms give an impression of stationarity but are not enough to decide. The precise measurement comes from unit root tests. In a model where the series is explained by its own lag, the test asks whether the coefficient equals 1:

If ρ = 1 the series has a unit root — it behaves like a random walk and is not stationary. The hypotheses of the Augmented Dickey-Fuller (ADF) test are:

nottem_random <- decompose(nottem)$random
nottem_random <- na.remove(nottem_random)  # NAs at both ends

tsdisplay(nottem_random)
adf.test(nottem_random)

	Augmented Dickey-Fuller Test

data:  nottem_random
Dickey-Fuller = -8.4454, Lag order = 6, p-value = 0.01
alternative hypothesis: stationary

The p-value is below 0.05, so H₀ is rejected: once the seasonal and trend components are removed, the series has no unit root and is stationary. We can move on to modelling.

7. AR, MA and ARMA

There are two basic ways to model a stationary series: explain it with its own past, or explain it with its past errors.

AR(p) — autoregressive process. Today's value is a linear combination of its own lagged values:

MA(q) — moving average process. Today's value is a weighted sum of past error terms:

ARMA(p,q). Most real series show both behaviours; the Box-Jenkins approach combines the two structures:

8. ARIMA(p,d,q)

ARMA works only on stationary series. For non-stationary series we difference first; adding this integration step, I(d), to the model gives ARIMA(p,d,q). For seasonal series a seasonal group of terms is added as well, and the model is written as ARIMA(p,d,q)(P,D,Q)[m], where m is the frequency.

Orders can also be chosen by hand. Below, three different models are fitted to nottem:

arima(nottem, order = c(2, 0, 0))
arima(nottem, order = c(0, 0, 3))
arima(nottem, order = c(2, 2, 2))

Of the three, ARIMA(2,0,0) has the lowest AIC. The logic is this: AIC accounts for both the fit to the data and the number of parameters, which keeps you out of the “a more complex model is always better” trap.

That ARIMA(2,2,2) performs worst is instructive too. nottem carries no non-stationary trend; differencing it twice adds unnecessary noise instead of fixing anything. Overdifferencing is a common mistake.

9. Choosing a Model with auto.arima()

When the number of series grows, or the data is too large to inspect by eye, trying orders one by one is not practical. auto.arima() from the forecast package determines d and D with unit root tests, then searches over p, q, P and Q using an information criterion.

auto.arima(nottem)

# To see the models that were tried
auto.arima(nottem, trace = TRUE)

# To turn off the stepwise search and search exhaustively
auto.arima(nottem, trace = TRUE, stepwise = FALSE)

Series: nottem
ARIMA(1,0,2)(1,1,2)[12] with drift

sigma^2 estimated as 5.22:  log likelihood=-516.48
AIC=1048.96   AICc=1049.62   BIC=1076.4

None of the models we tried by hand had a seasonal term; auto.arima() added the [12] seasonal component and a seasonal difference of D = 1. The error variance fell from 15.87 to 5.22 — seasonality accounts for most of the explained variation in this series.

With the stepwise search turned off (stepwise = FALSE) every combination of orders is tried and a slightly better model is found: ARIMA(1,0,1)(1,1,2)[12], AIC = 1046.49. The gain is small, while the search takes noticeably longer.

10. Forecasting and Interpretation

Once a model is fitted, forecast() produces predictions. Below, a 12-month forecast:

model <- auto.arima(nottem)
fc <- forecast(model, h = 12)

autoplot(fc) +
  labs(title = "nottem — 12-month ARIMA forecast",
       x = "Year", y = "Temperature (°F)")

The model captures the seasonal cycle: summer peaks and winter troughs appear in the right periods. But as the horizon lengthens, the forecasts move towards the mean of the series and the amplitude flattens slightly. This is not a flaw but ARIMA's natural behaviour — for the distant future, once the model has used up the information it has, it converges to the mean.

The practical takeaway: ARIMA is strong in the short run and should be read cautiously in the long run. The prediction intervals in the forecast() output say exactly the same thing; the bands widen as the horizon grows.

11. Common Mistakes

Conclusion

Time series analysis is a separate discipline for reading data that has an order. The flow is always similar: get to know the series, decompose it, check stationarity, use the correlograms to form an idea about the orders, fit the model, and finally interpret the forecast both statistically and in business terms.

The examples here ran on a single data set, but the logic does not change: electricity demand, units sold, mention volume or web traffic — the same questions apply to all of them. Which part of the series is trend, which part is seasonality, what is left over, and is what is left really random?

“A good time series model does not know the future; it is honest about how much uncertainty there is.”