Autocorrelation Function and Cross-correlation Function: Revealing the Correlations in Different Time Series

# Autocorrelation Function and Cross-correlation Function: Unveiling the Association Between Different Time Series

## 1. Concepts and Theoretical Foundations of Autocorrelation and Cross-correlation Functions

### 1.1 Autocorrelation Function

The Autocorrelation Function (ACF) measures the correlation between observations at specific time intervals within a time series. It is defined as:

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

where:

- `X_t` is the observation of the time series at time `t`
- `k` is the time interval
- `Cov()` is the covariance
- `Var()` is the variance

The range of values for the ACF is [-1, 1]. Positive values indicate positive correlation, negative values indicate negative correlation, and 0 indicates no correlation.

## 2. Calculation Methods for Autocorrelation and Cross-correlation Functions

### 2.1 Calculation of Autocorrelation Function

**Definition:**

The Autocorrelation Function (ACF) measures the correlation between an observation and itself at different time lags within a time series. It is defined as:

ρ(k) = Cov(X_t, X_{t+k}) / Var(X_t)

where:

- ρ(k) is the autocorrelation coefficient at lag k
- X_t is the observation of the time series at time t
- Cov(X_t, X_{t+k}) is the covariance between X_t and X_{t+k}
- Var(X_t) is the variance of X_t

**Calculation Method:**

The autocorrelation function can be calculated using the following steps:

1. Calculate the mean of the time series: μ = (1/N) ΣX_t
2. Calculate the variance of the time series: σ^2 = (1/N) Σ(X_t - μ)^2
3. Calculate the covariance at different lags k: Cov(X_t, X_{t+k}) = (1/N) Σ(X_t - μ)(X_{t+k} - μ)
4. Calculate the autocorrelation coefficient: ρ(k) = Cov(X_t, X_{t+k}) / σ^2

**Code Block:**

import numpy as np

def autocorr(x, k):
     """Calculate the autocorrelation function.

     Parameters:
         x: Time series
         k: Lag

     Returns:
         The autocorrelation coefficient at lag k
     """
     mean = np.mean(x)
     var = np.var(x)
     cov = np.cov(x, np.roll(x, k))[0, 1]
     return cov / var

**Logical Analysis:**

This code block implements the calculation of the autocorrelation function. It first calculates the mean and variance of the time series. Then, it uses the `np.cov()` function to calculate the covariance of the time series with itself at lag k. Finally, it divides the covariance by the variance to obtain the autocorrelation coefficient.

### 2.2 Calculation of Cross-correlation Function

**Definition:**

The Cross-correlation Function (CCF) measures the correlation between observations in two time series at different time lags. It is defined as:

ρ_{XY}(k) = Cov(X_t, Y_{t+k}) / (σ_X σ_Y)

where:

- ρ_{XY}(k) is the cross-correlation coefficient at lag k
- X_t is the observation of time series X at time t
- Y_{t+k} is the observation of time series Y at time t+k
- Cov(X_t, Y_{t+k}) is the covariance between X_t and Y_{t+k}
- σ_X is the standard deviation of time series X
- σ_Y is the standard deviation of time series Y

**Calculation Method:**

The cross-correlation function can be calculated using the following steps:

1. Calculate the means of the two time series: μ_X = (1/N) ΣX_t, μ_Y = (1/N) ΣY_t
2. Calculate the standard deviations of the two time series: σ_X = (1/N) Σ(X_t - μ_X)^2, σ_Y = (1/N) Σ(Y_t - μ_Y)^2
3. Calculate the covariance at different lags k: Cov(X_t, Y_{t+k}) = (1/N) Σ(X_t - μ_X)(Y_{t+k} - μ_Y)
4. Calculate the cross-correlation coefficient: ρ_{XY}(k) = Cov(X_t, Y_{t+k}) / (σ_X σ_Y)

**Code Block:**

import numpy as np

def crosscorr(x, y, k):
     """Calculate the cross-correlation function.

     Parameters:
         x: Time series X
         y: Time series Y
         k: Lag

     Returns:
         The cross-correlation coefficient at lag k
     """
     mean_x = np.mean(x)
     mean_y = np.mean(y)
     std_x = np.std(x)
     std_y = np.std(y)
     cov = np.cov(x, np.roll(y, k))[0, 1]
     return cov / (std_x * std_y)

**Logical Analysis:**

This code block implements the calculation of the cross-correlation function. I