# `RollingKurt`

## Description

The `RollingKurt` class computes the excess kurtosis of a data sequence within a specified moving window. This rolling calculation provides a measure of the "tailedness" of the data distribution over the window, with a correction applied for small sample sizes. The computed value represents excess kurtosis, meaning it is adjusted to measure how the distribution deviates from a normal distribution (where excess kurtosis is zero). Additionally, a bias correction (or sample correction) is included, making this estimate more accurate when sample sizes are small.

*Parameters*: 
- **`window_size`**: Specifies the size of the rolling window.
- **`start_policy`**: Defines how the function handles the initial phase when fewer than `window_size` data points are available. This parameter accepts one of the following three values:
  - `"strict"`: Returns `NaN` for all calculations until `window_size` elements have been processed.
  - `"expanding"`: Adapts the computation by dynamically reducing the window size to include all available data, starting from a single point and growing until `window_size` is reached.
  - `"zero"`: Simulates a full initial window of zeros, effectively pre-filling the data stream with `window_size` zeros before processing the actual input.

## Usage Example and Plot
Below is an example of using `RollingKurt` to calculate the rolling median for a random dataset, along with a plot illustrating its output.

```{eval-rst}
.. plotly::
    :include-source: True

    import numpy as np
    import plotly.graph_objects as go
    from plotly.subplots import make_subplots
    from screamer import RollingKurt

    # Generate example data
    data = np.cumsum(np.random.normal(size=300))

    # Create subplots with specified row heights and shared x-axis
    fig = make_subplots(
        rows=2, cols=1,
        shared_xaxes=True,
        row_heights=[2/3, 1/3],
        vertical_spacing=0.1
    )

    # Add traces for each subplot
    fig.add_trace(go.Scatter(y=data, mode='lines', name='Input Data'), row=1, col=1)
    fig.add_trace(go.Scatter(y=RollingKurt(30)(data), mode='lines', name='Rolling Kurtosis', line=dict(color='red')), row=2, col=1)

    # Update layout with titles and axis labels
    fig.update_layout(
        title=f"Rolling Kurtosis with Window Size 30",
        xaxis_title="Index",
        yaxis=dict(title="Input Data"),
        yaxis2=dict(title="Rolling Kurtosis", range=[-2, 4]),
        margin=dict(l=20, r=20, t=80, b=20),
        legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1)        
    )

    fig.show()
```


## Implementation Details

### Algorithm

`RollingKurt` implements cyclic buffers to accumulate windowed statistics.

### Complexity

* **Time Complexity**: `O(log(1))` per new element due to the insertion and deletion operations in the heaps.
* **Space Complexity**: `O(window_size)`, as only elements within the current window are stored.


### Performance

* Short streams (n=1.000): 120% faster than `Pandas Rolling kurt`
* Longer streams (n=1.000.000): 400% faster than `Pandas Rolling kurt`