Variance

Variance of input or sequence of inputs

Library:
DSP System Toolbox / Statistics

Description

The Variance block computes the unbiased variance of each row or column of the input, or along vectors of a specified dimension of the input. It can also compute the variance of the entire input. You can specify the dimension using the Find the variance value over parameter. The Variance block can also track the variance in a sequence of inputs over a period of time. To track the variance in a sequence of inputs, select the Running variance parameter.

Note

The Running mode in the Variance block will be removed in a future release. To compute the running variance in Simulink^®, use the Moving Variance block instead.

Ports

Input

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`In` — Data input
vector | matrix | N-D array

The block accepts real-valued or complex-valued multichannel and multidimensional inputs.

This port is unnamed until you select the Running variance parameter and set the Reset port parameter to any option other than None.

`Rst` — Reset port
scalar

Specify the event that causes the block to reset the running variance. The sample time of the Rst input must be a positive integer multiple of the input sample time.

Dependencies

To enable this port, select the Running variance parameter and set the Reset port parameter to any option other than None.

Output

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`Port_1` — Variance along the specified dimension
scalar | vector | matrix | N-D array

When you do not select the Running variance parameter, the block computes the variance in each row or column of the input, or along vectors of a specified dimension of the input. It can also compute the variance of the entire input at each individual sample time. Each element in the output array y is the variance of the corresponding column, row, or entire input. The output array y depends on the setting of the Find the variance value over parameter.

Consider a three-dimensional input signal of size M-by-N-by-P. When you set Find the variance value over to:

Entire input — The output at each sample time is a scalar that contains the variance of the M-by-N-by-P input matrix.
Each row — The output at each sample time consists of an M-by-1-by-P array, where each element contains the variance of each vector over the second dimension of the input. For an M-by-N matrix input, the output at each sample time is an M-by-1 column vector.
Each column — The output at each sample time consists of a 1-by-N-by-P array, where each element contains the variance of each vector over the first dimension of the input. For an M-by-N matrix input, the output at each sample time is a 1-by-N row vector.
In this mode, the block treats length-M unoriented vector inputs as M-by-1 column vectors.
Specified dimension — The output at each sample time depends on the value of the Dimension parameter. If you set the Dimension to 1, the output is the same as when you select Each column. If you set the Dimension to 2, the output is the same as when you select Each row. If you set the Dimension to 3, the output at each sample time is an M-by-N matrix containing the variance of each vector over the third dimension of the input.

When you select Running variance, the block tracks the variance of each channel in a time sequence of inputs. In this mode, you must also specify a value for the Input processing parameter. When you set Input processing to:

Elements as channels (sample based) — The block treats each element of the input as a separate channel. For a three-dimensional input signal of size M-by-N-by-P, the block outputs an M-by-N-by-P array. Each element y_ijk of the output contains the variance of the element u_ijk for all inputs since the last reset.
When a reset event occurs, the running variance y_ijk in the current frame is reset to the element u_ijk.
Columns as channels (frame based) — The block treats each column of the input as a separate channel. This option does not support input signals with more than two dimensions. For a two-dimensional input signal of size M-by-N, the block outputs an M-by-N matrix. Each element y_ij of the output contains the variance of the elements in the jth column of all inputs since the last reset, up to and including the element u_ij of the current input.
When a reset event occurs, the running variance for each channel becomes the variance of all the samples in the current input frame, up to and including the current input sample.

The data type of the output matches the data type of the input.

Parameters

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Main Tab

`Running variance` — Option to select running variance
off (default) | on

When you select the Running variance parameter, the block tracks the variance value of each channel in a time sequence of inputs.

`Find the variance value over` — Dimension over which variance is computed
`Each column` (default) | `Entire input` | `Each row` | `Specified dimension`

Each column — The block outputs the variance over each column.
Each row — The block outputs the variance over each row.
Entire input — The block outputs the variance over the entire input.
Specified dimension — The block outputs the variance over the dimension specified in the Dimension parameter.

Dependencies

To enable this parameter, clear the Running variance parameter.

`Dimension` — Custom dimension
`1` (default) | scalar

Specify the dimension (one-based value) of the input signal over which the variance is computed. The value of this parameter must be greater than 0 and less than or equal to the number of dimensions in the input signal.

Dependencies

To enable this parameter, set Find the variance value over to Specified dimension.

`Input processing` — Method to process the input in running mode
`Columns as channels (frame based)` (default) | `Elements as channels (sample based)`

Columns as channels (frame based) — The block treats each column of the input as a separate channel. This option does not support input signals with more than two dimensions. For a two-dimensional input signal of size M-by-N, the block outputs an M-by-N matrix. Each element y_ij of the output contains the variance of the elements in the jth column of all inputs since the last reset, up to and including the element u_ij of the current input.
When a reset event occurs, the running variance for each channel becomes the variance of all the samples in the current input frame, up to and including the current input sample.
Elements as channels (sample based) — The block treats each element of the input as a separate channel. For a three-dimensional input signal of size M-by-N-by-P, the block outputs an M-by-N-by-P array. Each element y_ijk of the output contains the variance of the element u_ijk for all inputs since the last reset.
When a reset event occurs, the running variance y_ijk in the current frame is reset to the element u_ijk.
Variable-Size Inputs
When your inputs are of variable size, and you select the Running variance parameter, then:
- If you set the Input processing parameter to Elements as channels (sample based), the state is reset.
- If you set the Input processing parameter to Columns as channels (frame based), then:
  - When the input size difference is in the number of channels (number of columns), the state is reset.
  - When the input size difference is in the length of channels (number of rows), the state is not reset, and the running operation is carried out as usual.

Dependencies

To enable this parameter, select the Running variance parameter.

`Reset port` — Reset event
`None` (default) | `Rising edge` | `Falling edge` | `Either edge` | `Non-zero sample`

The block resets the running variance whenever a reset event is detected at the optional Rst port. The reset sample time must be a positive integer multiple of the input sample time.

When a reset event occurs while the Input processing parameter is set to Elements as channels (sample based), the running variance for each channel is initialized to the value in the corresponding channel of the current input. Similarly, when the Input processing parameter is set to Columns as channels (frame based), the running variance for each channel becomes the variance of all the samples in the current input frame, up to and including the current input sample.

Use this parameter to specify the reset event.

None — Disables the Rst port.
Rising edge — Triggers a reset operation when the Rst input does one of the following:
- Rises from a negative value to either a positive value or zero.
- Rises from zero to a positive value, where the rise is not a continuation of a rise from a negative value to zero.
Falling edge — Triggers a reset operation when the Rst input does one of the following:
- Falls from a positive value to a negative value or zero.
- Falls from zero to a negative value, where the fall is not a continuation of a fall from a positive value to zero.
Either edge — Triggers a reset operation when the Rst input is a Rising edge or Falling edge.
Non-zero sample — Triggers a reset operation at each sample time, when the Rst input is not zero.

Note

When running simulations in the Simulink multitasking mode, reset signals have a one-sample latency. Therefore, when the block detects a reset event, there is a one-sample delay at the reset port rate before the block applies the reset. For more information on latency and the Simulink tasking modes, see Excess Algorithmic Delay (Tasking Latency) and Time-Based Scheduling and Code Generation (Simulink Coder).

Dependencies

To enable this parameter, select the Running variance parameter.

Data Types Tab

Note

To use these parameters, the data input must be fixed point. For all other inputs, the parameters on the Data Types tab are ignored.

`Rounding mode` — Method of rounding operation
`Floor` (default) | `Ceiling` | `Convergent` | `Nearest` | `Round` | `Simplest` | `Zero`

Specify the rounding mode for fixed-point operations. For more details, see rounding mode.

`Saturate on integer overflow` — Method of overflow action
off (default) | on

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on saturate and wrap, see overflow mode for fixed-point operations.

`Input-squared product output` — Data type of the input-squared term
`Same as input` (default) | `Binary point scaling`

The squares of the input elements are stored in the Input-squared product output data type. If the input is complex, the squares of the real and imaginary parts of the input are stored in this data type. For more details, see Fixed Point.

You can set this parameter to:

Inherit: Same as input — The data type is same as the input data type.
Binary point scaling — The Input-squared product output data type uses binary point scaling. If you select this option, the block displays the fields to specify the Word length and Fraction length. The Signedness is inherited from the input.

`Input-sum-squared product` — Data type of the input-sum-squared term
`Same as input-squared product` (default) | `Binary point scaling`

The squares of the sum of the input elements are stored in the Input-sum-squared product data type. If the input is complex, the squares of the sum of the real parts and the squares of the sum of the imaginary parts are stored in this data type. For more details, see Fixed Point.

You can set this parameter to:

Same as input-squared product — The data type is the same as the input squared-product data type.
Binary point scaling — The Input-sum-squared product data type uses binary point scaling. If you select this option, the block displays the fields to specify the Word length and Fraction length. The Signedness is inherited from the input.

`Accumulator` — Accumulator data type
`Same as input-squared product` (default) | `Same as input` | `Binary point scaling`

Accumulator specifies the data type of the output of an accumulation operation in the Variance block. See Fixed Point for illustrations depicting the use of the accumulator data type in this block.

You can set this parameter to:

Same as input-squared product — The accumulator data type is the same as the input-squared product data type.
Same as input — The accumulator data type is the same as the input data type.
Binary point scaling — The Accumulator data type uses binary point scaling. If you select this option, the block displays the fields to specify the Word length and Fraction length. The Signedness is inherited from the input.

`Output` — Output data type
`Same as input-squared product` (default) | `Same as accumulator` | `Same as input` | `Binary point scaling`

Output specifies the data type of the output of the Variance block. See Fixed Point for information about the use of the output data type in this block. You can set it to:

Same as input-squared product — The output data type is the same as the input-squared product data type.
Same as accumulator — The output data type is the same as the accumulator data type.
Same as input — The output data type is the same as the input data type.
Binary point scaling — The Output data type uses binary point scaling. If you select this option, the block displays the fields to specify the Word length and Fraction length. The Signedness is inherited from the input.

`Lock data type settings against changes by the fixed-point tools` — Prevent fixed-point tools from overriding data types
off (default) | on

Select this parameter to prevent the fixed-point tools from overriding the data types you specify on the block.

Block Characteristics

Data Types	`double` \| `fixed point` \| `integer` \| `single`
Direct Feedthrough	`no`
Multidimensional Signals	`no`
Variable-Size Signals	`yes`
Zero-Crossing Detection	`no`

More About

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Variance

The variance of a discrete-time signal is the square of the standard deviation of the signal. Variance gives a measure of deviation of the signal from its mean value.

For purely real or imaginary input, u, of size M-by-N, the variance is given by:

$y = σ^{2} = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {| u_{i j} |}^{2} - \frac{{| \sum_{i = 1}^{M} \sum_{j = 1}^{N} u_{i j} |}^{2}}{M * N}}{M * N - 1} .$

where,

u_ij is the input data element at indices i, j.
M is the length of the jth column.
N is the number of columns.

For complex inputs, the variance is given by the following equation:

$σ^{2} = σ_{Re}^{2} + σ_{Im}^{2}$

where,

σ_Re² is the variance of the real part of the complex input.
σ_Im² is the variance of the imaginary part of the complex input.

Algorithms

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Variance

When you clear the Running variance parameter in the block and specify a dimension, the block produces results identical to the MATLAB^® var function when it is called as y = var(u,0,D), where,

u is the data input.
D is the dimension.
y is the variance along the specified dimension.

When this block calculates the variance along the entire input, the result is identical to calling the var function as y = var(u(:)).

For a complex input signal, the variance is the sum of the variances of the real and imaginary parts.

$σ^{2} = σ_{Re}^{2} + σ_{Im}^{2}$

Extended Capabilities

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

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

For purely real or imaginary input u of size M-by-N, the variance is given by:

$y = σ^{2} = \frac{\sum_{i = 1}^{M} \sum_{j = 1}^{N} {| u_{i j} |}^{2} - \frac{{| \sum_{i = 1}^{M} \sum_{j = 1}^{N} u_{i j} |}^{2}}{M * N}}{M * N - 1} .$

The following diagram shows the data types used within the Variance block when the input is fixed-point.

For complex inputs, the variance is given by the following equation:

$σ^{2} = σ_{Re}^{2} + σ_{Im}^{2}$

Documentation

Variance

Description

Ports

Input

In — Data input vector | matrix | N-D array

Rst — Reset port scalar

Dependencies

Output

Port_1 — Variance along the specified dimension scalar | vector | matrix | N-D array

Parameters

Main Tab

Running variance — Option to select running variance off (default) | on

Find the variance value over — Dimension over which variance is computed Each column (default) | Entire input | Each row | Specified dimension

Dependencies

Dimension — Custom dimension 1 (default) | scalar

Dependencies

Input processing — Method to process the input in running mode Columns as channels (frame based) (default) | Elements as channels (sample based)

Dependencies

Reset port — Reset event None (default) | Rising edge | Falling edge | Either edge | Non-zero sample

Dependencies

Data Types Tab

Rounding mode — Method of rounding operation Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Saturate on integer overflow — Method of overflow action off (default) | on

Input-squared product output — Data type of the input-squared term Same as input (default) | Binary point scaling

Input-sum-squared product — Data type of the input-sum-squared term Same as input-squared product (default) | Binary point scaling

Accumulator — Accumulator data type Same as input-squared product (default) | Same as input | Binary point scaling

Output — Output data type Same as input-squared product (default) | Same as accumulator | Same as input | Binary point scaling

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types off (default) | on

Block Characteristics

More About

Variance

Algorithms

Variance

Extended Capabilities

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

Fixed-Point Conversion Design and simulate fixed-point systems using Fixed-Point Designer™.

See Also

Functions

Objects

Blocks

DSP System Toolbox Documentation

Support

`In` — Data input
vector | matrix | N-D array

`Rst` — Reset port
scalar

`Port_1` — Variance along the specified dimension
scalar | vector | matrix | N-D array

`Running variance` — Option to select running variance
off (default) | on

`Find the variance value over` — Dimension over which variance is computed
`Each column` (default) | `Entire input` | `Each row` | `Specified dimension`

`Dimension` — Custom dimension
`1` (default) | scalar

`Input processing` — Method to process the input in running mode
`Columns as channels (frame based)` (default) | `Elements as channels (sample based)`

`Reset port` — Reset event
`None` (default) | `Rising edge` | `Falling edge` | `Either edge` | `Non-zero sample`

`Rounding mode` — Method of rounding operation
`Floor` (default) | `Ceiling` | `Convergent` | `Nearest` | `Round` | `Simplest` | `Zero`

`Saturate on integer overflow` — Method of overflow action
off (default) | on

`Input-squared product output` — Data type of the input-squared term
`Same as input` (default) | `Binary point scaling`

`Input-sum-squared product` — Data type of the input-sum-squared term
`Same as input-squared product` (default) | `Binary point scaling`

`Accumulator` — Accumulator data type
`Same as input-squared product` (default) | `Same as input` | `Binary point scaling`

`Output` — Output data type
`Same as input-squared product` (default) | `Same as accumulator` | `Same as input` | `Binary point scaling`

`Lock data type settings against changes by the fixed-point tools` — Prevent fixed-point tools from overriding data types
off (default) | on

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

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.