dsp.MovingVariance

Moving variance

Description

The dsp.MovingVariance System object™ computes the moving variance of the input signal along each channel, independently over time. The object uses either the sliding window method or the exponential weighting method to compute the moving variance. In the sliding window method, a window of specified length is moved over the data, sample by sample, and the variance is computed over the data in the window. In the exponential weighting method, the object subtracts each sample of the data from the average, squares the difference, and multiplies the squared result with a weighting factor. The object then computes the variance by adding all the weighted data. For more details on these methods, see Algorithms.

To compute the moving variance of the input:

Create the dsp.MovingVariance object and set its properties.
Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?.

Creation

Syntax

MovVar = dsp.MovingVariance

MovVar = dsp.MovingVariance(Len)

MovVar = dsp.MovingVariance(Name,Value)

Description

MovVar = dsp.MovingVariance returns a moving variance object, MovVar, using the default properties.

example

MovVar = dsp.MovingVariance(Len) sets the WindowLength property to Len.

example

MovVar = dsp.MovingVariance(Name,Value) specifies additional properties using Name,Value pairs. Unspecified properties have default values.

Example:

MovVar = dsp.MovingVariance('Method','Exponential
            weighting','ForgettingFactor',0.9);

Properties

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Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`Method` — Method to compute the variance
`'Sliding window'` (default) | `'Exponential weighting'`

'Sliding window' — A window of length specified by SpecifyWindowLength is moved over the input data along each channel. For every sample the window moves by, the object computes the variance over the data in the window.
'Exponential weighting' — The object subtracts each sample of the data from the average, squares the difference, and multiplies the squared result with a weighting factor. The object then computes the variance by adding all the weighted data. The magnitude of the weighting factors decreases exponentially as the age of the data increases, never reaching zero.

For more details on these methods, see Algorithms.

`SpecifyWindowLength` — Specify window length
`true` (default) | `false`

Flag to specify a window length, specified as a scalar Boolean.

true — The length of the sliding window is equal to the value you specify in the WindowLength property.
false — The length of the sliding window is infinite. In this mode, the variance is computed using the current sample and all past samples.

Dependencies

This property applies when you set Method to 'Sliding window'.

`WindowLength` — Length of the sliding window
4 (default) | positive scalar integer

Length of the sliding window in samples, specified as a positive scalar integer.

Dependencies

This property applies when you set Method to 'Sliding window' and SpecifyWindowLength to true.

`ForgettingFactor` — Exponential weighting factor
0.9 (default) | positive real scalar in the range (0,1]

Exponential weighting factor, specified as a positive real scalar in the range (0,1].

A forgetting factor of 0.9 gives more weight to the older data than does a forgetting factor of 0.1. A forgetting factor of 1.0 indicates infinite memory. All the past samples are given an equal weight.

Since this property is tunable, you can change its value even when the object is locked.

Tunable: Yes

Dependencies

This property applies when you set Method to 'Exponential weighting'.

Data Types: single | double

Usage

Syntax

y = movVar(x)

Description

example

y = movVar(x) computes the moving variance of the input signal, x, using either the sliding window method or exponential weighting method.

Input Arguments

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`x` — Data input
vector | matrix

Data input, specified as a vector or a matrix. If x is a matrix, each column is treated as an independent channel. The moving variance is computed along each channel. The object also accepts variable-size inputs. Once the object is locked, you can change the size of each input channel, but cannot change the number of channels.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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`y` — Moving variance
vector | matrix

Moving variance of the input signal, returned as a vector or a matrix.

Data Types: single | double
Complex Number Support: Yes

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

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Common to All System Objects

`step`	Run System object algorithm
`release`	Release resources and allow changes to System object property values and input characteristics
`reset`	Reset internal states of System object

Examples

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Moving Variance of Noisy Square Wave Signal

Open Script

Compute the moving variance of a noisy square wave signal with varying amplitude using the dsp.MovingVariance object.

Initialization

Set up movvarWindow and movvarExp objects. movvarWindow uses the sliding window method with a window length of 800. movvarExp uses the exponentially weighting method with a forgetting factor of 0.999. Create a time scope for viewing the output.

FrameLength = 100;
Fs = 100;
movvarWindow = dsp.MovingVariance(800);
movvarExp = dsp.MovingVariance('Method','Exponential weighting',...
    'ForgettingFactor',0.999);
scope  = timescope('SampleRate',Fs,...
    'TimeSpanOverrunAction','Scroll',...
    'TimeSpanSource','Property',...
    'TimeSpan',1000,...
    'ShowGrid',true,...
    'BufferLength',1e7,...
    'YLimits',[0 7e-4]);
title = 'Actual (yellow) Sliding Window (blue) Exponentially Weighted (red) variance';
scope.Title = title;

Compute the Variance

Generate a noisy square wave signal. Vary the amplitude of the square wave after a given number of frames. Apply the sliding window method and the exponentially weighting method on this signal. The actual variance is np. This value is used while adding noise to the data. Compare the actual variance with the computed variances on the time scope.

count = 1;
noisepower = 1e-4 * [1 2 3 4];
index = 1;
for index = 1:length(noisepower)
    np = noisepower(index);
    yexp = np*ones(FrameLength,1);
    for i = 1:250
        x = 1 + sqrt(np) * randn(FrameLength,1);
        y1 = movvarWindow(x);
        y2 = movvarExp(x);
        scope([yexp,y1,y2]);
    end
end

Algorithms

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Sliding Window Method

In the sliding window method, the output at the current sample is the variance of the current sample with respect to the data in the window. To compute the first Len – 1 outputs, when the window does not have enough data yet, the algorithm fills the window with zeros. As an example, to compute the variance when the second input sample comes in, the algorithm fills the window with Len – 2 zeros. Len is the length of the window in samples. The data vector, x, is then the two data samples followed by Len – 2 zeros.

When you do not specify the window length, the algorithm chooses an infinite window length. In this mode, the output is the moving variance of the current sample with respect to all previous samples in the channel.

Consider an example of computing the moving variance of a streaming input data using the sliding window method. The algorithm uses a window length of 4. With each input sample that comes in, the window of length 4 moves along the data.

Exponential Weighting Method

In the exponential weighting method, the moving variance is computed recursively using these formulas:

$\begin{array}{l} s^{2}_{N, λ} = \frac{1}{v_{N, λ}} \sum_{k = 1}^{N} λ^{N - k} {[x_{k} - {\bar{x}}_{N, λ}]}^{2} \\ v_{N, λ} = \frac{2 λ (1 - λ^{N - 1})}{(1 - λ) (1 + λ)} \end{array}$

To compute the moving variance, the algorithm implements these equations recursively.

$s^{2}_{N, λ}$ — Moving variance of the current data sample with respect to the rest of the data in the channel.
${\bar{x}}_{N, λ}$ — Moving average at the current sample. For details on computing the moving average, see dsp.MovingAverage.
${[x_{k} - {\bar{x}}_{N, λ}]}^{2}$ — Difference between each data sample and the average of the data, squared.
$\sum_{k = 1}^{N} λ^{N - k} {[x_{k} - {\bar{x}}_{N, λ}]}^{2}$ — Difference between each data sample and the average of the data, squared and multiplied with the forgetting factor. All the squared terms are added.
$\frac{1}{v_{N, λ}}$ — Weighting factor applied to the sum.
λ — Forgetting factor you can specify through the ForgettingFactor property.

As the age of the data increases, the magnitude of the weighting factor decreases exponentially, and never reaches zero. In other words, the recent data has more influence on the current variance, than the older data.

The value of the forgetting factor determines the rate of change of the weighting factors. A forgetting factor of 0.9 gives more weight to the older data than does a forgetting factor of 0.1. A forgetting factor of 1.0 indicates infinite memory. All the past samples are given an equal weight.

Consider an example of computing the moving variance using the exponential weighting method. The forgetting factor is 0.9.

References

[1] Bodenham, Dean. “Adaptive Filtering and Change Detection for Streaming Data.” PH.D. Thesis. Imperial College, London, 2012.

Documentation

dsp.MovingVariance

Description

Creation

Syntax

Description

Properties

`Method` — Method to compute the variance
`'Sliding window'` (default) | `'Exponential weighting'`

`SpecifyWindowLength` — Specify window length
`true` (default) | `false`

Dependencies

`WindowLength` — Length of the sliding window
4 (default) | positive scalar integer

Dependencies

`ForgettingFactor` — Exponential weighting factor
0.9 (default) | positive real scalar in the range (0,1]

Dependencies

Usage

Syntax

Description

Input Arguments

`x` — Data input
vector | matrix

Output Arguments

`y` — Moving variance
vector | matrix

Object Functions

Common to All System Objects

Examples

Moving Variance of Noisy Square Wave Signal

Algorithms

Sliding Window Method

Exponential Weighting Method

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

See Also

Objects

Blocks

Topics

DSP System Toolbox Documentation

Support

Documentation

dsp.MovingVariance

Description

Creation

Syntax

Description

Properties

Method — Method to compute the variance 'Sliding window' (default) | 'Exponential weighting'

SpecifyWindowLength — Specify window length true (default) | false

Dependencies

WindowLength — Length of the sliding window 4 (default) | positive scalar integer

Dependencies

ForgettingFactor — Exponential weighting factor 0.9 (default) | positive real scalar in the range (0,1]

Dependencies

Usage

Syntax

Description

Input Arguments

x — Data input vector | matrix

Output Arguments

y — Moving variance vector | matrix

Object Functions

Common to All System Objects

Examples

Moving Variance of Noisy Square Wave Signal

Algorithms

Sliding Window Method

Exponential Weighting Method

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

See Also

Objects

Blocks

Topics

DSP System Toolbox Documentation

Support

`Method` — Method to compute the variance
`'Sliding window'` (default) | `'Exponential weighting'`

`SpecifyWindowLength` — Specify window length
`true` (default) | `false`

`WindowLength` — Length of the sliding window
4 (default) | positive scalar integer

`ForgettingFactor` — Exponential weighting factor
0.9 (default) | positive real scalar in the range (0,1]

`x` — Data input
vector | matrix

`y` — Moving variance
vector | matrix

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.