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Udemy - Learn Matlab x. Matlab ra Linux Cracked thepiratebay Digital Signal Processing with Matlab Examples [] thepiratebay Using Matlab kickass. Fundamental Chemistry with Matlab kickass. Training kickass. 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: lowess uses a linear polynomial, while loess uses a quadratic polynomial. 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. The local regression smoothing process follows these steps for each data point:. 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:. The data point to be smoothed has the largest weight and the most influence on the fit.
Data points outside the span have zero weight and no influence on the fit. 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. Note that unlike the moving average smoothing process, the span never changes. For example, when you smooth the data point with the smallest predictor value, the shape of the weight function is truncated by one half, the leftmost data point in the span has the largest weight, and all the neighboring points are to the right of the smoothed value.
The weight function for an end point and for an interior point is shown below for a span of 31 data points. Using the lowess method with a span of five, the smoothed values and associated regressions for the first four data points of a generated data set are shown below. Notice that the span does not change as the smoothing process progresses from data point to data point.
However, depending on the number of nearest neighbors, the regression weight function might not be symmetric about the data point to be smoothed. In particular, plots a and b use an asymmetric weight function, while plots c and d use a symmetric weight function. For the loess method, the graphs would look the same except the smoothed value would be generated by a second-degree polynomial. If your data contains outliers, the smoothed values can become distorted, and not reflect the behavior of the bulk of the neighboring data points.
To overcome this problem, you can smooth the data using a robust procedure that is not influenced by a small fraction of outliers. For a description of outliers, refer to Residual Analysis. Curve Fitting Toolbox software provides a robust version for both the lowess and loess smoothing methods.
These robust methods include an additional calculation of robust weights, which is resistant to outliers. The robust smoothing procedure follows these steps:. Calculate the residuals from the smoothing procedure described in the previous section.
Compute the robust weights for each data point in the span. The weights are given by the bisquare function,. The median absolute deviation is a measure of how spread out the residuals are. If r i is small compared to 6 MAD , then the robust weight is close to 1. If r i is greater than 6 MAD , the robust weight is 0 and the associated data point is excluded from the smooth calculation.
Smooth the data again using the robust weights. The final smoothed value is calculated using both the local regression weight and the robust weight. Repeat the previous two steps for a total of five iterations. The smoothing results of the lowess procedure are compared below to the results of the robust lowess procedure for a generated data set that contains a single outlier. The span for both procedures is 11 data points.
Plot a shows that the outlier influences the smoothed value for several nearest neighbors. Plot b suggests that the residual of the outlier is greater than six median absolute deviations. Therefore, the robust weight is zero for this data point. Plot c shows that the smoothed values neighboring the outlier reflect the bulk of the data.
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