Moving Variance

Moving variance

• Library:

• DSP System Toolbox / Statistics

Description

The Moving Variance block computes the moving variance of the input signal along each channel independently over time. The block 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 moves over the data sample by sample, and the block computes the variance over the data in the window. In the exponential weighting method, the block subtracts each sample of the data from the average, squares the difference, and multiplies the squared result by a weighting factor. The block then computes the variance by adding all the weighted data. For more details on these methods, see Algorithms.

Ports

Input

x — Data input

column vector | row vector | matrix

Data over which the block computes the moving variance. The block accepts real-valued or complex-valued multichannel inputs, that is, m-by-n size inputs, where m ≥ 1, and n ≥ 1. The block also accepts variable-size inputs. During simulation, you can change the size of each input channel. However, the number of channels cannot change.

This port is unnamed until you set Method to Exponential weighting and select the Specify forgetting factor from input port parameter.

Data Types: single | double
Complex Number Support: Yes

lambda — Forgetting factor

positive real scalar in the range (0,1]

The forgetting factor determines how much weight past data is given. 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 previous samples are given an equal weight.

Dependencies

This port appears when you set Method to Exponential weighting and select the Specify forgetting factor from input port parameter.

Data Types: single | double

Output

Port_1 — Moving variance output

column vector | row vector | matrix

The size of the moving variance output matches the size of the input. The block uses either the sliding window method or the exponential weighting method to compute the moving variance, as specified by the Method parameter. For more details, see Algorithms.

Data Types: single | double
Complex Number Support: Yes

Parameters

If a parameter is listed as tunable, then you can change its value during simulation.

Method — Moving variance method

Sliding window (default) | Exponential weighting

• Sliding window — A window of length Window length moves over the input data along each channel. For every sample the window moves over, the block computes the variance over the data in the window.

• Exponential weighting — The block subtracts each sample of the data from the average, squares the difference, and multiplies the squared result by a weighting factor. The block 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, but the magnitude never reaches zero.

For more details on these methods, see Algorithms.

Specify window length — Flag to specify window length

on (default) | off

When you select this check box, the length of the sliding window is equal to the value you specify in Window length. When you clear this check box, the length of the sliding window is infinite. In this mode, the block computes the variance of the current sample with respect to all the previous samples in the channel.

Dependencies

This parameter appears when you set Method to Sliding window.

Window length — Length of sliding window

4 (default) | positive scalar integer

Specifies the length of the sliding window.

Dependencies

This parameter appears when you set Method to Sliding window and select the Specify window length check box.

Specify forgetting factor from input port — Flag to specify forgetting factor

off (default) | on

When you select this check box, the forgetting factor is input through the lambda port. When you clear this check box, the forgetting factor is specified on the block dialog through the Forgetting factor parameter.

Dependencies

This parameter appears only when you set Method to Exponential weighting.

Forgetting factor — Exponential weighting factor

0.9 (default) | positive real scalar in the range (0,1]

The forgetting factor determines how much weight past data is given. 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 previous samples are given an equal weight.

Tunable: Yes

Dependencies

This parameter appears when you set Method to Exponential weighting and clear the Specify forgetting factor from input port check box.

Simulate using — Type of simulation to run

Code generation (default) | Interpreted execution

• Code generation

  Simulate model using generated C code. The first time you run a simulation, Simulink^® generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time but provides faster simulation speed than Interpreted execution.

• Interpreted execution

  Simulate model using the MATLAB^®  interpreter. This option shortens startup time but has slower simulation speed than Code generation.

Block Characteristics

Data Types	double | single
Multidimensional Signals	No
Variable-Size Signals	Yes

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 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. 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:

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

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.

Extended Capabilities

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