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A Savitzky–Golay filter is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the signal-to-noise ratio without greatly distorting the signal. This is achieved, in a process known as convolution, by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares. When the data points are equally spaced an analytical solution to the least-squares equations can be found, in the form of a single set of "convolution coefficients" that can be applied to all data sub-sets, to give estimates of the smoothed signal, (or derivatives of the smoothed signal) at the central point of each sub-set. The method, based on established mathematical procedures,〔. "Graduation Formulae obtained by fitting a Polynomial."〕 was popularized by Abraham Savitzky and Marcel J. E. Golay who published tables of convolution coefficients for various polynomials and sub-set sizes in 1964. Some errors in the tables have been corrected. The method has been extended for the treatment of 2- and 3-dimensional data. Savitzky and Golay's paper is one of the most widely cited papers in the journal ''Analytical Chemistry'' and is classed by that journal as one of its "10 seminal papers" saying "it can be argued that the dawn of the computer-controlled analytical instrument can be traced to this article". == Applications == The data consists of a set of ''n'' points (''j'' = 1, ..., ''n''), where ''x'' is an independent variable and ''y''''j'' is an observed value. They are treated with a set of ''m'' convolution coefficients, ''C''i, according to the expression : It is easy to apply this formula in a spreadsheet. Selected convolution coefficients are shown in the tables, below. For example, for smoothing by a 5-point quadratic polynomial, ''m'' = 5, ''i'' = −2, −1, 0, 1, 2 and the ''j''th smoothed data point, ''Y''j, is given by :, where, ''C''−2 = −3/35, ''C''−1 = 12 / 35, etc. There are numerous applications of smoothing, which is performed primarily to make the data appear to be less noisy than it really is. The following are applications of numerical differentiation of data. Note When calculating the ''n''th. derivative an additional scaling factor of may be applied to all calculated data points to obtain absolute values (see expressions for , below, for details). #Location of maxima and minima in experimental data curves. This was the application that first motivated Savitzky.〔 The first derivative of a function is zero at a maximum or minimum. The diagram shows data points belonging to a synthetic Lorentzian curve, with added noise (blue diamonds). Data are plotted on a scale of half width, relative to the peak maximum at zero. The smoothed curve (red line) and 1st. derivative (green) were calculated with 7-point cubic Savitzky–Golay filters. Linear interpolation of the first derivative values at positions either side of the zero-crossing gives the position of the peak maximum. 3rd. derivatives can also be used for this purpose. #Location of an end-point in a titration curve. An end-point is an inflection point where the second derivative of the function is zero. The titration curve for malonic acid illustrates the power of the method. The first end-point at 4 ml is barely visible, but the second derivative allows its value to be easily determined by linear interpolation to find the zero crossing. #Baseline flattening. In analytical chemistry it is sometimes necessary to measure the height of an absorption band against a curved baseline. Because the curvature of the baseline is much less than the curvature of the absorption band, the second derivative effectively flattens the baseline. Three measures of the derivative height, which is proportional to the absorption band height, are the "peak-to-valley" distances h1 and h2, and the height from baseline, h3. #Resolution enhancement in spectroscopy. Bands in the second derivative of a spectroscopic curve are narrower than the bands in the spectrum: they have reduced half-width. This allows partially overlapping bands to be "resolved" into separate (negative) peaks. The diagram illustrates how this may be used also for chemical analysis, using measurement of "peak-to-valley" distances. In this case the valleys are a property of the 2nd. derivative of a Lorentzian. (''x''-axis position is relative to the position of the peak maximum on a scale of half width at half height). #Resolution enhancement with 4th. derivative (positive peaks). The minima are a property of the 4th derivative of a Lorentzian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Savitzky–Golay filter」の詳細全文を読む スポンサード リンク
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