Autocorrelation Function and Fourier Transform: Unveiling the Frequency Characteristics of Time Series Data

# 1. Introduction to Time Series Analysis

Time series analysis is a statistical method used to study data sequences that change over time. It is widely applied in fields such as finance, economics, and meteorology. Time series data exhibit the following characteristics:

- **Orderliness:** Data is arranged in chronological order with time being the independent variable.
- **Correlation:** There is a correlation between adjacent data points, allowing past values to predict future ones.
- **Stationarity:** Statistical properties of the time series (such as mean and variance) remain relatively stable over a period.

# 2. Autocorrelation Function

### 2.1 Definition and Properties of the Autocorrelation Function

#### 2.1.1 Mathematical Expression of the Autocorrelation Function

The autocorrelation function (ACF) measures the correlation between observations at a certain time lag within a time series. For a time series $\{X_t\}$, the ACF is defined as:

ACF(k) = Cov(X_t, X_{t+k}) / Var(X_t)

Where:

* $k$ is the time lag.
* $Cov(X_t, X_{t+k})$ is the covariance between observations at times $t$ and $t+k$.
* $Var(X_t)$ is the variance of the observation at time $t$.

#### 2.1.2 Properties and Applications of the Autocorrelation Function

The autocorrelation function has the following properties:

***Symmetry:** $ACF(k) = ACF(-k)$
***Normalization:** $0 \leq ACF(k) \leq 1$
***Maximum Value of 1:** $ACF(0) = 1$

The shape of the ACF can provide important information about the characteristics of the time series:

***Positive Autocorrelation:** $ACF(k) > 0$ indicates that observations are correlated over time, i.e., adjacent observations tend to have similar values.
***Negative Autocorrelation:** $ACF(k) < 0$ indicates that observations are negatively correlated over time, i.e., adjacent observations tend to have opposite values.
***Periodicity:** The periodicity of the ACF suggests the presence of seasonal or cyclical patterns within the time series.

### 2.2 Applications of the Autocorrelation Function in Time Series Analysis

#### 2.2.1 Identifying Trends and Seasonality in Time Series

The autocorrelation function can help identify trends and seasonality in a time series.

***Trend:** If the ACF remains positive over long lags, it suggests a trend in the time series.
***Seasonality:** If the ACF shows periodic peaks at specific lags, it indicates seasonality in the time series.

#### 2.2.2 Testing for Time Series Stationarity

Stationarity of a time series refers to its statistical properties (such as mean, variance, and autocorrelation function) remaining constant over time. The ACF can be used to test for time series stationarity:

***Stationary Time Series:** The ACF is close to zero for all lags.
***Non-stationary Time Series:** The ACF is significantly non-zero at certain lags, indicating a trend or seasonality in the time series.

**Example:**

Consider the following time series:

X = [10, 12, 14, 16, 18, 20, 22, 24, 26, 28]

Calculate the autocorrelation function as follows:

| k | ACF(k) |
|---|---|
| 0 | 1.000 |
| 1 | 0.900 |
| 2 | 0.700 |
| 3 | 0.500 |
| 4 | 0.300 |
| 5 | 0.100 |
| 6 | -0.10