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数据平滑和离群值检测

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数据平滑指用于消除数据中不需要的噪声或行为的技术，而离群值检测用于标识与其余数据显著不同的数据点。

移动窗方法

移动窗方法是分批处理数据的方式，通常是为了从统计角度表示数据中的相邻点。移动平均值是一种常见的数据平滑技术，它沿着数据滑动窗，同时计算每个窗内点的均值。这可以帮助消除从一个数据点到下一个数据点的非显著变化。

例如，假设每分钟测量一次风速，持续约 3 小时。使用 movmean 函数和 5 分钟的窗口大小可去除高速阵风。

load windData.mat
mins = 1:length(speed);
window = 5;
meanspeed = movmean(speed,window);
plot(mins,speed,mins,meanspeed)
axis tight
legend("Measured Wind Speed","Average Wind Speed over 5 min Window")
xlabel("Time")
ylabel("Speed")

Figure contains an axes object. The axes object with xlabel Time, ylabel Speed contains 2 objects of type line. These objects represent Measured Wind Speed, Average Wind Speed over 5 min Window.

同样，您可以使用 movmedian 函数计算滑动窗中的风速中位数。

medianspeed = movmedian(speed,window);
plot(mins,speed,mins,medianspeed)
axis tight
legend("Measured Wind Speed","Median Wind Speed over 5 min Window")
xlabel("Time")
ylabel("Speed")

Figure contains an axes object. The axes object with xlabel Time, ylabel Speed contains 2 objects of type line. These objects represent Measured Wind Speed, Median Wind Speed over 5 min Window.

并非所有数据都适合用移动窗方法进行平滑处理。例如，创建插入了随机噪声的正弦信号。

t = 1:0.2:15;
A = sin(2*pi*t) + cos(2*pi*0.5*t);
Anoise = A + 0.5*rand(1,length(t));
plot(t,A,t,Anoise)
axis tight
legend("Original Data","Noisy Data")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Noisy Data.

使用移动均值和大小为 3 的窗口对含噪数据进行平滑处理。

window = 3;
Amean = movmean(Anoise,window);
plot(t,A,t,Amean)
axis tight
legend("Original Data","Moving Mean - Window Size 3")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Moving Mean - Window Size 3.

移动均值方法可获得数据的大致形状，但不能非常准确地捕获波谷（局部最小值）。由于波谷点在每个窗口中两个较大的邻点之间，因此均值不是那些点的理想近似值。如果使窗口大小变大，均值将完全消除较短的波峰。对于这种类型的数据，您可能需要考虑其他平滑技术。 

Amean = movmean(Anoise,5);
plot(t,A,t,Amean)
axis tight
legend("Original Data","Moving Mean - Window Size 5")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Moving Mean - Window Size 5.

常见的平滑方法

smoothdata 函数提供几种平滑选项，如萨维茨基-戈雷方法，这是一种常用的信号处理平滑技术。默认情况下，smoothdata 根据数据为方法选择最佳估计窗口大小。

使用萨维茨基-戈雷方法可对含噪信号 Anoise 进行平滑处理，并输出它使用的窗口大小。与 movmean 相比，该方法可提供更好的波谷近似值。 

[Asgolay,window] = smoothdata(Anoise,"sgolay");
plot(t,A,t,Asgolay)
axis tight
legend("Original Data","Savitzky-Golay","location","best")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Savitzky-Golay.

window
window = 3

稳健的 Lowess 方法是另一种平滑方法，尤其适用于含噪数据还包含离群值的情形。在含噪数据中插入离群值，并使用稳健的 Lowess 方法对数据进行平滑处理，从而消除离群值。 

Anoise(36) = 20;
Arlowess = smoothdata(Anoise,"rlowess",5);
plot(t,Anoise,t,Arlowess)
axis tight
legend("Noisy Data","Robust Lowess")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Noisy Data, Robust Lowess.

检测离群值

数据中的离群值可能使数据处理结果和其他计算量严重失真。例如，如果您尝试用移动平均值方法对包含离群值的数据进行平滑处理，则可能得到误导性的波峰或波谷。 

Amedian = smoothdata(Anoise,"movmedian");
plot(t,Anoise,t,Amedian)
axis tight
legend("Noisy Data","Moving Median")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Noisy Data, Moving Median.

当检测到离群值时，isoutlier 函数返回逻辑值 1。验证 Anoise 中离群值的索引和值。

TF = isoutlier(Anoise);
ind = find(TF)
ind = 36

Aoutlier = Anoise(ind)
Aoutlier = 20

您可以使用 filloutliers 函数并指定填充方法来替换数据中的离群值。例如，用紧挨 Anoise 中离群值右侧的邻点值填充该离群值。

Afill = filloutliers(Anoise,"next");
plot(t,Anoise,t,Afill,"o-")
axis tight
legend("Noisy Data with Outlier","Noisy Data with Filled Outlier")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Noisy Data with Outlier, Noisy Data with Filled Outlier.

您也可以使用 rmoutliers 函数从数据中删除离群值。例如，删除 Anoise 中的离群值。

Aremove = rmoutliers(Anoise);
plot(t,Anoise,t(~TF),Aremove,"o-")
axis tight
legend("Noisy Data with Outlier","Noisy Data with Outlier Removed")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Noisy Data with Outlier, Noisy Data with Outlier Removed.

非均匀数据

并非所有数据都由等间距点组成，这会影响数据处理的方法。创建一个 datetime 向量，其中包含 Airreg 中数据的不规则采样时间。time 向量表示了前 30 分钟内每分钟采集一次的样本和两天内每小时采集一次的样本。 

t0 = datetime(2014,1,1,1,1,1);
timeminutes = sort(t0 + minutes(1:30));
timehours = t0 + hours(1:48);
time = [timeminutes timehours];
Airreg = rand(1,length(time));
plot(time,Airreg)
axis tight

Figure contains an axes object. The axes object contains an object of type line.

默认情况下，smoothdata 按照等间距整数进行平滑处理，在本例中为 1,2,...,78。由于整数时间戳与 Airreg 中各点的采样不协调，前半小时的数据在平滑后仍然出现噪声。 

Adefault = smoothdata(Airreg,"movmean",3);
plot(time,Airreg,time,Adefault)
axis tight
legend("Original Data","Smoothed Data with Default Sample Points")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Smoothed Data with Default Sample Points.

MATLAB® 中的许多数据处理函数（包括 smoothdata、movmean 和 filloutliers）允许您提供样本点，从而确保相对于其采样单位和频率处理数据。要消除 Airreg 中前半小时数据的高频变化，请将 SamplePoints 名称-值参量和 time 中的时间戳结合使用。

Asamplepoints = smoothdata(Airreg,"movmean", ...
    hours(3),"SamplePoints",time);
plot(time,Airreg,time,Asamplepoints)
axis tight
legend("Original Data","Smoothed Data with Sample Points")

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Data, Smoothed Data with Sample Points.

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