Compute the **autocorrelation function (ACF)** for a time series to measure how similar the series is to a lagged version of itself. You’ll implement a simple, beginner-friendly ACF that returns correlations for lags from 0 up to a given maximum lag.

The autocorrelation at lag $k$ is defined as:
$$
\rho(k)=\frac{\sum_{t=0}^{n-k-1}(x_t-\mu)(x_{t+k}-\mu)}{\sum_{t=0}^{n-1}(x_t-\mu)^2}
$$

## Requirements

Implement the function
```python
def autocorrelation(x, max_lag):
    \"""
    Computes the autocorrelation function (ACF) up to max_lag.

    Args:
        x (np.ndarray): Time series values of length n.
        max_lag (int): Maximum lag k to compute (inclusive).

    Returns:
        np.ndarray: A 1D array of length (max_lag + 1) where the k-th entry is rho(k).
    \"""
    # Your solution below
```

**Rules:**
- Return ACF values for all integer lags `k = 0, 1, ..., max_lag`.
- Use the same mean `mu` computed from the full series `x` for every lag.
- Use the denominator $\sum_{t=0}^{n-1}(x_t-\mu)^2$ (so `rho(0)` should be `1.0`).
- Do not call any prebuilt autocorrelation/correlation utilities (e.g., `np.correlate` or `pd.Series.autocorr`).
- Use vectorized NumPy operations (e.g. slicing) for calculating the sums, rather than Python loops over time steps.
- Return the result as a NumPy array.

## Example

```python
x = np.array([1.0, 2.0, 1.0, 2.0])
max_lag = 2

autocorrelation(x, max_lag)
```

Output:
```python
[1.0, -0.75, 0.5]
```

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Autocorrelation Function

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1507. Autocorrelation Function

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Similar Questions

Argument	Type
x	np.ndarray
max_lag	int