THRESHOLD

This function applies wavelet-style thresholding to each value in an input range. Values are modified according to a selected thresholding rule and a threshold magnitude.

In soft thresholding, values are shrunk toward zero by the threshold after sign preservation, while in hard thresholding values below the threshold are replaced by a substitute value. Additional modes include garrote, greater, and less thresholding rules.

For soft thresholding, an input value x and threshold \lambda produce:

\hat{x} = \operatorname{sign}(x)\max(|x|-\lambda, 0)

This enables denoising workflows by attenuating small-magnitude coefficients.

Excel Usage

=THRESHOLD(data, value, threshold_mode, substitute)
  • data (list[list], required): Input numeric values as an Excel range.
  • value (float, required): Threshold value.
  • threshold_mode (str, optional, default: “soft”): Thresholding mode.
  • substitute (float, optional, default: 0): Replacement value for thresholded entries.

Returns (list[list]): Thresholded values in a 2D range matching the input shape.

Example 1: Soft threshold on a single row

Inputs:

data value threshold_mode substitute
1 1.5 2 2.5 3 2 soft 0

Excel formula:

=THRESHOLD({1,1.5,2,2.5,3}, 2, "soft", 0)

Expected output:

Result
0 0 0 0.5 1
Example 2: Hard threshold on a 2x3 matrix

Inputs:

data value threshold_mode substitute
1 2 3 2 hard 0
4 0.5 2.5

Excel formula:

=THRESHOLD({1,2,3;4,0.5,2.5}, 2, "hard", 0)

Expected output:

Result
0 2 3
4 0 2.5
Example 3: Greater mode replaces values below threshold

Inputs:

data value threshold_mode substitute
1 2 3 4 2.5 greater -1

Excel formula:

=THRESHOLD({1,2,3,4}, 2.5, "greater", -1)

Expected output:

Result
-1 -1 3 4
Example 4: Less mode replaces values above threshold

Inputs:

data value threshold_mode substitute
1 2 3 4 2.5 less 9

Excel formula:

=THRESHOLD({1,2,3,4}, 2.5, "less", 9)

Expected output:

Result
1 2 9 9

Python Code

Show Code 
import numpy as np
import pywt

def threshold(data, value, threshold_mode='soft', substitute=0):
    """
    Apply elementwise wavelet thresholding to numeric data.

    See: https://pywavelets.readthedocs.io/en/latest/ref/thresholding-functions.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        data (list[list]): Input numeric values as an Excel range.
        value (float): Threshold value.
        threshold_mode (str, optional): Thresholding mode. Valid options: Soft, Hard, Garrote, Greater, Less. Default is 'soft'.
        substitute (float, optional): Replacement value for thresholded entries. Default is 0.

    Returns:
        list[list]: Thresholded values in a 2D range matching the input shape.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        data = to2d(data)
        if not isinstance(data, list) or not all(isinstance(row, list) for row in data):
            return "Error: Invalid input - data must be a 2D list"

        matrix = []
        for row in data:
            out_row = []
            for item in row:
                try:
                    out_row.append(float(item))
                except (TypeError, ValueError):
                    return "Error: Input must contain only numeric values"
            matrix.append(out_row)

        arr = np.asarray(matrix, dtype=float)
        result = pywt.threshold(arr, value=float(value), mode=threshold_mode, substitute=float(substitute))
        return np.asarray(result).tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

Input numeric values as an Excel range.
Threshold value.
Thresholding mode.
Replacement value for thresholded entries.