`Diff2`#

Description#

The Diff2 function computes the second-order finite difference of the input - the discrete analogue of a second derivative. It is the result of applying a one-step difference twice.

Equation:

\[ y[t] = x[t] - 2 \, x[t-1] + x[t-2] \]

equivalent to \(\Delta(\Delta x)\).

Parameters:

start_policy (str, optional): warmup behaviour, "strict" (default), "truncate", or "zero". Two warmup samples are needed before a valid output is produced.

NaN handling: Under start_policy="strict" the first two outputs are NaN. NaN inputs propagate.

Note: Diff2 is not the same as Diff(2). Diff(2) is the lag-2 first difference \(x[t] - x[t-2]\). Diff2 is the second-order difference, i.e. the difference of differences. On a quadratic input \(x[t] = a t^2 + b t + c\) the result is the constant \(2a\).

NaN handling#

Policy: propagate. Input NaN values are stored in the lookback. Output is NaN at any index where the function's positional formula references a NaN input; recovery happens once the NaN slides out of the lookback.

Examples#

Usage example#

import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots
from screamer import Diff2

n = 200
t = np.arange(n)
x = 0.0005 * (t - 100)**2 + np.sin(t / 10.0)
out = Diff2()(x)

fig = make_subplots(rows=2, cols=1, shared_xaxes=True,
                    row_heights=[0.5, 0.5], vertical_spacing=0.1)
fig.add_trace(go.Scatter(y=x, mode='lines', name='Input (parabola + sine)'),
              row=1, col=1)
fig.add_trace(go.Scatter(y=out, mode='lines',
                         name='Diff2 (discrete second derivative)',
                         line=dict(color='green')),
              row=2, col=1)
fig.update_layout(
    title="Diff2: Discrete Second Derivative",
    xaxis_title="Index",
    yaxis_title="Original",
    yaxis2_title="Diff2",
    margin=dict(l=20, r=20, t=60, b=20),
    legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1)
)
fig.show()

Reference#

Equivalent to applying numpy.diff twice (with a warmup of two NaN samples preserved).

Diff2#