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Filtering and Smoothing Data

For this reason, a Savitzky-Golay filter is also called a digital smoothing polynomial filter or a least-squares smoothing filter. Note that a higher degree polynomial makes it possible to achieve a high level of smoothing without attenuation of data features.

The Savitzky-Golay filtering method is often used with frequency data or with spectroscopic peak data. For frequency data, the method is effective at preserving the high-frequency components of the signal. For spectroscopic data, the method is effective at preserving higher moments of the peak such as the line width.

By comparison, the moving average filter tends to filter out a significant portion of the signal's high-frequency content, and it can only preserve the lower moments of a peak such as the centroid. However, Savitzky-Golay filtering can be less successful than a moving average filter at rejecting noise. Normally, Savitzky-Golay filtering requires uniform spacing of the predictor data.

However, the Curve Fitting Toolbox algorithm supports nonuniform spacing. Therefore, you are not required to perform an additional filtering step to create data with uniform spacing. The plot shown below displays generated Gaussian data and several attempts at smoothing using the Savitzky-Golay method. The data is very noisy and the peak widths vary from broad to narrow.

Plot a shows the noisy data. To more easily compare the smoothed results, plots b and c show the data without the added noise. Plot b shows the result of smoothing with a quadratic polynomial. Notice that the method performs poorly for the narrow peaks. Plot c shows the result of smoothing with a quartic polynomial.

In general, higher degree polynomials can more accurately capture the heights and widths of narrow peaks, but can do poorly at smoothing wider peaks. The smoothing process is considered local because, like the moving average method, each smoothed value is determined by neighboring data points defined within the span. The process is weighted because a regression weight function is defined for the data points contained within the span. In addition to the regression weight function, you can use a robust weight function, which makes the process resistant to outliers.

Finally, the methods are differentiated by the model used in the regression: The local regression smoothing methods used by Curve Fitting Toolbox software follow these rules:. You can specify the span as a percentage of the total number of data points in the data set. For example, a span of 0. Compute the regression weights for each data point in the span. The weights are given by the tricube function shown below.

The weights have these characteristics:. A weighted linear least-squares regression is performed. For lowess, the regression uses a first degree polynomial.

For loess, the regression uses a second degree polynomial. The smoothed value is given by the weighted regression at the predictor value of interest. If the smooth calculation involves the same number of neighboring data points on either side of the smoothed data point, the weight function is symmetric. However, if the number of neighboring points is not symmetric about the smoothed data point, then the weight function is not symmetric. Henri de Montaut - http: Charles Ortel, "Sunday with Charles.

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Clean Beto, Biden and Bernie are the frontline of the Democratic hopefuls for Results are returned in the column vector yy. The default span for the moving average is 5. Because of the way endpoints are handled, the result differs from the result returned by the filter function. Supported values for method are listed in the table below. A lowpass filter with filter coefficients equal to the reciprocal of the span. A generalized moving average with filter coefficients determined by an unweighted linear least-squares regression and a polynomial model of specified degree default is 2.

The method can accept nonuniform predictor data. A robust version of 'lowess' that assigns lower weight to outliers in the regression. The method assigns zero weight to data outside six mean absolute deviations. A robust version of 'loess' that assigns lower weight to outliers in the regression. For the loess and lowess methods, span is a percentage of the total number of data points, less than or equal to 1.

For the moving average and Savitzky-Golay methods, span must be odd an even span is automatically reduced by 1. You should specify x data when it is not uniformly spaced or sorted. If x is not uniform and you do not specify method , lowess is used. If the smoothing method requires x to be sorted, the sorting occurs automatically.

The input gpuarrayY is a gpuArray column vector. The output gpuarrayYY is a gpuArray column vector.