Matlab数据处理之移动方差movvar函数

Matlab中，可以使用movvar函数获取数组的移动方差，可以理解为局部 k 个数据点的方差值组成的数组。本文将从以下几个方面介绍movvar函数：movvar函数常见用法、movvar函数语法说明、movvar函数实例。其中，movvar函数实例包括：向量的中心移动方差、向量的尾部移动方差、指定移动方差的归一化、矩阵的移动方差、包含 NaN 元素的向量的移动方差、基于样本点计算移动方差、仅返回满窗口方差。

movvar函数帮助文档如下：

>> help movvar
 movvar   Moving variance value.
    Y = movvar(X,K) for a vector X and positive integer scalar K computes a
    centered moving variance by sliding a window of length K along X. Each
    element of Y is the local variance of the corresponding values of X
    inside the window, with Y the same size as X. When K is even, the
    window is centered about the current and previous elements of X. The
    sliding window is truncated at the endpoints where there are fewer than
    K elements from X to fill the window.
    
    For N-D arrays, movvar operates along the first array dimension whose
    size does not equal 1.
 
    By default, movvar normalizes by K-1 if K>1. If X consists of
    independent, identically distributed samples, then movvar is an
    unbiased estimator of the variance of the population of each window.
 
    Y = movvar(X,[NB NF]) for a vector X and nonnegative integers NB and NF
    computes a moving variance along the length of X, returning the local
    variance of the previous NB elements, the current element, and the next
    NF elements of X.
 
    movvar(X,K,NRM) specifies the normalization factor for the variance and
    can be one of the following:
 
        0   - (default) normalizes by K-1 for K>1 and by K when K=1.
        1   - normalizes by K and produces the second moment of the window
              about its mean.
 
    Y = movvar(X,K,NRM,DIM) or Y = movvar(X,[NB NF],NRM,DIM) operates along
    dimension DIM of X. When specifying DIM, you must specify NRM.
 
    movvar(...,MISSING) specifies how NaN (Not-a-Number) values are treated
    and can be one of the following:
 
        'includenan'   - (default) the variance of any window containing
                         NaN values is also NaN.
        'omitnan'      - the variance of any window containing NaN values
                         is the variance of all its non-NaN elements. If
                         all elements are NaN, the result is NaN.
 
    movvar(...,'Endpoints',ENDPT) controls how the variance is calculated
    at the endpoints of X, where there are not enough elements to fill the
    window. ENDPT can be either a scalar numeric or logical value or one of
    the following:
 
        'shrink'    - (default) compute the variance over the number of
                      elements of X that are inside the window, effectively
                      reducing the window size to fit X at the endpoints.
        'fill'      - compute the variance over the full window size,
                      filling missing values from X with NaN. This is
                      equivalent to padding X with NaN at the endpoints.
        'discard'   - compute the variance only when the window is filled
                      with elements of X, discarding partial endpoint
                      calculations and their corresponding elements in Y.
                      This truncates the output; for a vector X and window
                      length K, Y has length LENGTH(X)-K+1.
                      
    When ENDPT is a scalar numeric or logical value, the missing elements
    of X inside the window are replaced with that value and Y remains the
    same size as X.
 
    Example: Compute a 5-point centered moving variance.
        t = 1:10;
        x = [4 8 6 -1 -2 -3 -1 3 4 5];
        yc = movvar(x,5);
        plot(t,x,t,yc);
 
    Example: Compute a 5-point trailing moving variance.
        t = 1:10;
        x = [4 8 6 -1 -2 -3 -1 3 4 5];
        yt = movvar(x,[4 0]);
        plot(t,x,t,yt);
 
    Example: Compute a 5-point centered moving variance, padding the ends
    of the input with NaN.
        t = 1:10;
        x = [4 8 6 -1 -2 -3 -1 3 4 5];
        yp = movvar(x,5,'Endpoints','fill');
        plot(t,x,t,yp);
 
    Example: Compute a 5-point trailing moving variance, ignoring the first
    4 window shifts that do not contain 5 input elements.
        x = [4 8 6 -1 -2 -3 -1 3 4 5];
        yd = movvar(x,[4 0],'Endpoints','discard');

movvar函数常见用法

M = movvar(A,k)
M = movvar(A,[kb kf])
M = movvar(___,w)
M = movvar(___,w,dim)
M = movvar(___,nanflag)
M = movvar(___,Name,Value)

movvar函数语法说明

M = movvar(A,k) 返回由局部 k 个数据点的方差值组成的数组，其中每个方差基于 A 的相邻元素的长度为 k 的滑动窗计算得出。当 k 为奇数时，窗口以当前位置的元素为中心。当 k 为偶数时，窗口以当前元素及其前一个元素为中心。当没有足够的元素填满窗口时，窗口将自动在端点处截断。当窗口被截断时，只根据窗口内的元素计算方差。M 与 A 的大小相同。

如果 A 是向量，movvar 将沿向量 A 的长度运算。
如果 A 是多维数组，则 movvar 沿 A 的大小不等于 1 的第一个维度进行运算。

M = movvar(A,[kb kf]) 通过长度为 kb+kf+1 的窗口计算方差，其中包括当前位置的元素、前面的 kb 个元素和后面的 kf 个元素。

M = movvar(_,w) 为上述任意语法指定归一化因子。当 w = 0 时（默认值），M 按 k-1 对窗长度 k 进行归一化。当 w = 1 时，M 按 k 进行归一化。

M = movvar(_,w,dim) 为上述任一语法指定 A 的运算维度。指定 dim 时，始终在上述语法中指定权重 w。例如，如果 A 是矩阵，则 movvar(A,k,0,2) 沿 A 的列运算，计算每行的 k 个元素的移动方差。归一化因子是默认值 k-1。

M = movvar(_,nanflag) 指定在上述任意语法的计算中包括还是忽略 NaN 值。movvar(A,k,’includenan’) 会在计算中包括所有 NaN 值，而 movvar(A,k,’omitnan’) 则忽略这些值并基于较少的点计算方差。

M = movvar(_,Name,Value) 使用一个或多个名称-值对组参数指定方差的其他参数。例如，如果 x 是时间值向量，则 movvar(A,k,’SamplePoints’,x) 相对于 x 中的时间计算移动方差。

movvar函数实例

向量的中心移动方差

计算行向量的三点中心移动方差。当端点处的窗口中少于三个元素时，将根据可用元素计算方差。

>> A = [4 8 6 -1 -2 -3 -1 3 4 5];
>> M = movvar(A,3)

M =

    8.0000    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000    0.5000

向量的尾部移动方差

计算行向量的三点尾部移动方差。当端点处的窗口中少于三个元素时，将根据可用元素计算方差。

>> A = [4 8 6 -1 -2 -3 -1 3 4 5];
>> M = movvar(A,[2 0])

M =

         0    8.0000    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000

指定移动方差的归一化

计算行向量的三点中心移动方差，并按照窗口中的元素数对每个方差进行归一化。

>> A = [4 8 6 -1 -2 -3 -1 3 4 5];
>> M = movvar(A,3,1)

M =

    4.0000    2.6667   14.8889   12.6667    0.6667    0.6667    6.2222    4.6667    0.6667    0.2500

矩阵的移动方差

计算矩阵中每行的三点中心移动方差。窗口从第一行开始，沿水平方向移动到该行的末尾，然后移到第二行，依此类推。维度参数为 2，即跨 A 的列移动窗口。指定维度时，始终指定归一化因子。

>> A = [4 8 6; -1 -2 -3; -1 3 4];
>> M = movvar(A,3,0,2)

M =

    8.0000    4.0000    2.0000
    0.5000    1.0000    0.5000
    8.0000    7.0000    0.5000

包含 NaN 元素的向量的移动方差

计算包含两个 NaN 元素的行向量的三点中心移动方差。

>> A = [4 8 NaN -1 -2 -3 NaN 3 4 5];
>> M = movvar(A,3)

M =

    8.0000       NaN       NaN       NaN    1.0000       NaN       NaN       NaN    1.0000    0.5000

重新计算方差，但忽略 NaN 值。当 movvar 舍弃 NaN 元素时，它将根据窗口中的剩余元素计算方差。

>> M = movvar(A,3,'omitnan')

M =

    8.0000    8.0000   40.5000    0.5000    1.0000    0.5000   18.0000    0.5000    1.0000    0.5000

基于样本点计算移动方差

根据时间向量 t，计算 A 中数据的 3 小时中心移动方差。

>> A = [4 8 6 -1 -2 -3];
>> k = hours(3);
>> t = datetime(2016,1,1,0,0,0) + hours(0:5)

t = 

1 至 5 列

   2016-01-01 00:00:00   2016-01-01 01:00:00   2016-01-01 02:00:00   2016-01-01 03:00:00   2016-01-01 04:00:00

6 列

   2016-01-01 05:00:00

>> M = movvar(A,k,'SamplePoints',t)

M = 1×6

    8.0000    4.0000   22.3333   19.0000    1.0000    0.5000

仅返回满窗口方差

计算行向量的三点中心移动方差，但在输出中舍弃使用的点数少于三个的计算。也就是说，只返回从满的三元素窗口计算的方差，而舍弃端点计算。

>> A = [4 8 6 -1 -2 -3 -1 3 4 5];
>> M = movvar(A,3,'Endpoints','discard')

M =

    4.0000   22.3333   19.0000    1.0000    1.0000    9.3333    7.0000    1.0000