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.

The dsp.MovingVariance object and the movvar function both compute the moving variance of the input signal. However, the object can process large streams of real-time data and handle system states automatically. The function performs one-time computations on data that is readily available and cannot handle system states. For a comparison between the two, see System Objects vs MATLAB Functions.

To compute the moving variance of the input:

1. Create the dsp.MovingVariance object and set its properties.

2. 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(Len,Overlap)

MovVar = dsp.MovingVariance(Name,Value)

Description

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

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

MovVar = dsp.MovingVariance(Len,Overlap) sets the WindowLength property to Len and the OverlapLength property to Overlap.

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

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.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

OverlapLength — Overlap length between windows

WindowLength − 1 (default) | nonnegative integer

Overlap length between sliding windows, specified as a nonnegative integer. The value of overlap length varies in the range [0, WindowLength − 1]. If not specified, the overlap length is set to WindowLength − 1.

Dependencies

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

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

ForgettingFactor — Exponential weighting factor

0.9 (default) | non-negative real scalar in the range [0,1]

Exponential weighting factor, specified as a non-negative 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 and all the past samples are given an equal weight. A forgetting factor of 0 indicates no memory and the past samples have no weight on the current computation.

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

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

Input Arguments

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 accepts variable-size inputs. Once the object is locked, you can change the size of each input channel, but you cannot change the number of channels.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

y — Moving variance

vector | matrix

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

When you input a signal of size m-by-n to the object, and if you set Method to 'Sliding window' and SpecifyWindowLength to true, the output has an upper bound size of ceil(m/hop size)-by-n. Hop size is window length − overlap length. In other cases, the output has a size of m-by-n.

When you generate code from this object, the variable-size behavior of the output in the generated code depends on the input frame length and whether the size of the input signal is fixed or variable. For more details, see Code Generation.

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)

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

Moving Variance of Noisy Square Wave Signal

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

Initialization

Set up movvarWindow, movvarWindow_overlap, and movvarExp objects. movvarWindow uses the sliding window method with a window length of 800 samples and a default overlap length of 799 samples, which is 1 sample less than the specified window length. movvarWindow_overlap uses a window length of 800 samples and an overlap length of 700 samples. 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);
movvarWindow_overlap = dsp.MovingVariance(800,700);
movvarExp = dsp.MovingVariance('Method','Exponential weighting',...
    'ForgettingFactor',0.999);
scope  = timescope('SampleRate',[Fs,Fs,Fs/(800-700),Fs],...
    'TimeSpanOverrunAction','Scroll',...
    'TimeSpanSource','Property',...
    'TimeSpan',1000,...
    'ShowGrid',true,...
    'BufferLength',1e7,...
    'YLimits',[0 7e-4]);
title = 'Moving Variance';
scope.Title = title;
scope.ChannelNames = {'Original Signal',...
    'Sliding window of 800 samples with default overlap',...
    'Sliding window of 800 samples with an overlap of 700 samples',...
    'Exponential weighting with forgetting factor of 0.999'};

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];
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 = movvarWindow_overlap(x);
        y3 = movvarExp(x);
        scope(yexp,y1,y2,y3);
    end
end

Algorithms

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 output sample, the algorithm waits until it receives the hop size number of input samples. Hop size is defined as window length – overlap length. Remaining samples in the window are considered to be zero. As an example, if the window length is 5 and the overlap length is 2, then the algorithm waits until it receives 3 samples of input to compute the first sample of the output. After generating the first output, it generates the subsequent output samples for every hop size number of input samples.

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 and an overlap length of 3. 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:

$$\begin{array}{l}{s}^2{}_{N,\lambda }=\frac{1}{v_{N,\lambda }}{\displaystyle \sum _{k=1}^N{\lambda }^{N-k}}{\left[x_k-{\overline{x}}_{N,\lambda }\right]}^2\\ v_{N,\lambda }=\frac{2\lambda (1-{\lambda }^{N-1})}{(1-\lambda )(1+\lambda )}\end{array}$$

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

• $$s^2{}_{N,\lambda }$$ — Moving variance of the current data sample with respect to the rest of the data in the channel.

• $${\overline{x}}_{N,\lambda }$$ — Moving average at the current sample. For details on computing the moving average, see dsp.MovingAverage.

• $${\left[x_k-{\overline{x}}_{N,\lambda }\right]}^2$$ — Difference between each data sample and the average of the data, squared.

• $${\displaystyle \sum _{k=1}^N{\lambda }^{N-k}}{\left[x_k-{\overline{x}}_{N,\lambda }\right]}^2$$ — Difference between each data sample and the average of the data, squared and multiplied with the forgetting factor. All the squared terms are added.

• $$\frac{1}{v_{N,\lambda }}$$ — 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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• See System Objects in MATLAB Code Generation (MATLAB Coder).

• When you set Method to 'Sliding window' and SpecifyWindowLength to true, and generate code from this object, the variable-size behavior of the output depends on the input frame length and whether the size of the input signal is fixed or variable.

  See this table for more details.

  Input signal	Input size	Output signal
  Fixed-size	m-by-n	When the input frame length is a multiple of the hop size, the output signal has a fixed-size of (m/hop size)-by-n. When input frame length is not a multiple of the hop size, the output signal is variable-sized and has an upper bound of ceil(m/hop size)-by-n.
  Variable-size	m-by-n	Output is a variable-size signal. Output has an upper bound size of ceil(m/hop size)-by-n.

Version History

Introduced in R2016b

R2024b: Specify a forgetting factor of 0

You can now specify a forgetting factor of 0 in the dsp.MovingVariance object.

R2022b: Change in variable-size behavior for output signal in generated code

Starting in R2022b, if you generate code from this object with the Method property set to 'Sliding window' and the SpecifyWindowLength property set to true, and if you input a signal with a frame length that is a multiple of the hop size (window length − overlap length), this object generates a fixed-size output signal in the generated code. For more details, see Code Generation.