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| [[File:Lissage sg3 anim.gif|thumb|400px|Animation showing smoothing being applied, passing through the data from left to right. The red line represent the local polynomial being used to fit a sub-set of the data. The smoothed values are shown as circles.]] | | Just närvarande ledtråd tricks närheten . Vegas även 24 timmar av och till även fortsättningsvis icke med . Samt staden såsom icke - absolut där restauranger, barer, nattklubbar livsmedelsbutiker knappast nära. du varit på Vegas innan och betagen försöker något nytt kanske ni är mer frågvis [http://wordpress.org/search/i+kolla i kolla] in bakom kulisserna på synd Metropolis, äger kommit rätt område. Med odla avsevärt att handla samt åt en soluppgång att handla det, hurså tidrymd Vegas räkna ut baksida av underben [http://www.google.Co.uk/search?hl=en&gl=us&tbm=nws&q=sj%C3%A4lv+ska&gs_l=news själv ska] företa?<br><br> |
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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 (mathematics)|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,<ref name=Whittaker>{{cite book|last=Whittaker|first=E.T|title=The Calculus Of Observations|year=1924|publisher=Blackie And Son Limited|url=http://www.archive.org/details/calculusofobserv031400mbp|coauthors=Robinson,G|pages=291–296}}. "Graduation Formulae obtained by fitting a Polynomial."</ref><ref name=Guest>{{cite book|last=Guest|first=P.G.|title=Numerical Methods of Curve Fitting|year=1961|publisher=Cambridge University Press|chapter=Chapter 7, Estimation of Polynomial Coefficients}}</ref> 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.<ref name=SG>{{cite journal |first1=A. |last1=Savitzky |authorlink1=Abraham Savitzky |first2=M.J.E. |last2=Golay |authorlink2=Marcel J. E. Golay |year=1964 |title=Smoothing and Differentiation of Data by Simplified Least Squares Procedures |journal=[[Analytical Chemistry (journal)|Analytical Chemistry]] |volume=36 |issue=8 |pages=1627–1639 |doi=10.1021/ac60214a047}}</ref><ref name=SG2>{{cite journal|first1=Abraham|last1=Savitzky |year=1989 |title=A Historic Collaboration|journal=Analytical Chemistry|volume=61|issue=15 |pages=921A–923A |doi=10.1021/ac00190a744}}</ref> Some errors in the tables have been corrected.<ref name=Steinier>{{cite journal |first1=Jean|last1=Steinier |first2=Yves|last2=Termonia |first3=Jules |last3=Deltour |year=1972 |title=Smoothing and differentiation of data by simplified least square procedure|journal=Analytical Chemistry|volume=44 |issue=11 |pages=1906–1909 |doi=10.1021/ac60319a045}}</ref> The method has been extended for the treatment of 2- and 3-dimensional data.
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| Savitzky and Golay's paper is one of the most widely cited papers in the journal [[Analytical Chemistry (journal)|''Analytical Chemistry'']]<ref>{{cite journal |last1=Larive |first1= Cynthia K. |last2=Sweedler |first2=Jonathan V. |year=2013 |title=Celebrating the 75th Anniversary of the ACS Division of Analytical Chemistry: A Special Collection of the Most Highly Cited Analytical Chemistry Papers Published between 1938 and 2012 |journal=Analytical Chemistry |volume=85|issue=0 |pages= 4201–4202|doi=10.1021/ac401048d}}</ref> 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".<ref name='analytchemsem'>{{cite journal |first1=James |last1=Riordon |first2=Elizabeth |last2=Zubritsky |first3=Alan |last3=Newman |year=2000 |title=Top 10 Articles |journal=[[Analytical Chemistry (journal)|Analytical Chemistry]] |volume=72 |issue=9 |pages=24 A–329 A|url=http://pubs.acs.org/doi/pdf/10.1021/ac002801q |doi=10.1021/ac002801q }}</ref>
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| == Applications ==
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| The data consists of a set of ''n'' {''x''<sub>''j''</sub>, ''y''<sub>''j''</sub>} points (''j'' = 1, ..., ''n''), where ''x'' is an independent variable and ''y''<sub>''j''</sub> is an observed value. They are treated with a set of ''m'' convolution coefficients, ''C''<sub>i</sub> according to the expression
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| :<math>Y_j= \sum _{i=-(m-1)/2}^{i=(m-1)/2}C_i\, y_{j+i}\qquad \frac{m+1}{2} \le j \le n-\frac{m-1}{2}</math>
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| It is easy to apply this formula in a [[spreadsheet]]. Selected convolution coefficients are shown in the [[#Appendix|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''<sub>j</sub>, is given by
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| :<math>Y_j = \frac{1}{35} (-3 \times y_{j - 2} + 12 \times y_{j - 1} + 17 \times y_j + 12 \times y_{j + 1} -3 \times y_{j + 2})</math>,
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| where, ''C''<sub>−2</sub> = −3/35, ''C''<sub>−1</sub> = 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.<ref>{{cite book|last=Talsky|first=Gerhard|title=Derivative Spectrophotometry|publisher=Wiley|isbn=3527282947|url=http://books.google.co.uk/books?id=5dvvAAAAMAAJ&q=}}</ref>
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|
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| {{multiple image
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| |align=center
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| | image1 = Lorentzian and derivative.gif | |
| | width1 = 250
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| | caption1 = (1) Synthetic Lorentzian + noise (blue) and 1st. derivative (green)
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| | image2 = Malonic titration.png
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| | width2 = 250
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| | caption2 = (2) Titration curve (blue) for [[malonic acid]] and 2nd. derivative (green). The part in the light blue box is magnified 10 times | |
| | width3 = 250
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| | image3 = Baseline correction.png | |
| | caption3 = (3) Lorentzian on exponential baseline (blue) and 2nd. derivative (green) | |
| }}
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| {{multiple image
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| |align=center
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| | image1 = Resolution enhancement.png | |
| | width1 = 250
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| | caption1 =(4) Sum of two Lorentzians (blue) and 2nd. derivative (green) | |
| | image2 = 4th derivative.png
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| | width2 = 250
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| | caption2 = (5) 4th. derivative of the sum of two Lorentzians
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| }}
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| #Location of [[maxima and minima]] in experimental data curves. This was the application that first motivated Savitzky.<ref name=SG2/> The first derivative of a function is zero at a maximum or minimum. The diagram shows data points belonging to a synthetic [[Cauchy distribution|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. [[interpolation#linear interpolation|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.
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| #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.<ref>{{cite journal |last1=Abbaspour |first1=Abdolkarim |last2=Khajehzadeha |first2=Abdolreza |year=2012 |title=End point detection of precipitation titration by scanometry method without using indicator |journal=Anal. Methods |volume=4 |issue= |pages=923–932|doi=10.1039/C2AY05492B}}</ref> The titration curve for [[malonic acid]] illustrates the power of the method. The first [[Equivalence point|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.
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| #Baseline flattening. In [[analytical chemistry]] it is sometimes necessary to measure the height of an [[absorption band]] against a curved baseline.<ref>{{cite journal |last1=Li |first1=N |last2=Li |first2=XY |last3=Zou |first3=XZ |last4=Lin |first4=LR |last5=Li |first5=YQ
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| |year=2011 |title=A novel baseline-correction method for standard addition based derivative spectra and its application to quantitative analysis of benzo(a)pyrene in vegetable oil samples. |journal=Analyst |volume=136 |issue=13 |pages=2802–2810|doi=10.1039/c0an00751j}}</ref> 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.<ref>{{cite journal|last=Dixit|first=L.|coauthors=Ram, S.|title=Quantitative Analysis by Derivative Electronic Spectroscopy|journal=Applied Spectroscopy Reviews|year=1985|volume=21|issue=4|pages=311–418|doi=10.1080/05704928508060434}}</ref>
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| #Resolution enhancement in spectroscopy. Bands in the second derivative of a spectroscopic curve are narrower than the bands in the spectrum: they have reduced [[Full width at half maximum|half-width]]. This allows partially overlapping bands to be "resolved" into separate (negative) peaks.<ref>{{cite journal |first1=Arthur T. |last1=Giese |first2=C. Stacey |last2=French|journal=Appl. Spectrosc. |year=1955 |title=The Analysis of Overlapping Spectral Absorption Bands by Derivative Spectrophotometry|volume=9 |issue=2 |pages=78–96 }}</ref> 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 [[Full width at half maximum|half width at half height]]).
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| #Resolution enhancement with 4th. derivative (positive peaks). The minima are a property of the 4th derivative of a Lorentzian.
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| == Derivation of convolution coefficients ==
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| When the data points are equally spaced an [[analytical solution]] to the least-squares equations can be found.<ref name=Guest/> This solution forms the basis of the [[convolution]] method of numerical smoothing and differentiation. Suppose that the data consists of a set of ''n'' {''x''<sub>''j''</sub>, ''y''<sub>''j''</sub>} points (''j'' = 1, ..., ''n''), where ''x'' is an independent variable and ''y''<sub>''j''</sub> is a datum value. A polynomial will be fitted by [[linear least squares (mathematics)|linear least squares]] to a set of ''m'' (an odd number) adjacent data points, each separated by an interval ''h''. Firstly, a change of variable is made
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| :<math>z = {{x - \bar x} \over h}</math>
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| where <math>{\bar x}</math> is the value of the central point. ''z'' takes the values <math>{1-m\over2},\ldots,0,\ldots, {m-1\over2}</math> (e.g. ''m'' = 5 → ''z'' = −2, −1, 0, 1, 2).<ref group=note>With even values of ''m'', ''z'' will run from 1 − ''m'' to ''m'' − 1 in steps of 2</ref> The polynomial, of degree ''k'' is defined as
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| :<math>Y = a_0 + a_1 z + a_2 z^2 \cdots + a_k z^k.</math><ref group=note>The simple [[moving average]] is a special case with ''k'' = 0, ''Y'' = ''a''<sub>0</sub>. In this case all convolution coefficients are equal to 1/''m''.</ref>
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| The coefficients ''a''<sub>0</sub>, ''a''<sub>1</sub> etc. are obtained by solving the [[normal equations]]
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| :<math>{\mathbf{a}} = \left( {{\mathbf{J}}^{\mathbf{T}} {\mathbf{J}}} \right)^{ - {\mathbf{1}}} {\mathbf{J}}^{\mathbf{T}} {\mathbf{y}}</math>
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| where the ''i''th row of the [[Jacobian matrix]], <math>\mathbf J =\mathbf{{\partial Y\over \partial a}}</math>, has values 1, ''z''<sub>i</sub>, ''z''<sub>i</sub><sup>2</sup>, .. .
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| For example, for a cubic polynomial fitted to 5 points, ''z''= −2, −1, 0, 1, 2 the normal equations are solved as follows.
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| :<math>
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| \mathbf{J} =
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| \begin{pmatrix}
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| 1 & -2 & 4 & -8 \\
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| 1 & - 1 & 1 & -1 \\
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| 1 & 0 & 0 & 0 \\
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| 1 & 1 & 1 & 1 \\
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| 1 & 2 & 4 & 8
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| \end{pmatrix}
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| </math> | |
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| :<math>
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| \mathbf{J^TJ} =
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| \begin{pmatrix} m & \sum z & \sum z^2 & \sum z^3 \\
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| \sum z & \sum z^2 & \sum z^3 & \sum z^4 \\
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| \sum z^2 & \sum z^3 & \sum z^4 & \sum z^5 \\
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| \sum z^3 & \sum z^4 & \sum z^5 & \sum z^6 \\ | |
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| \end{pmatrix}=
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| \begin{pmatrix} m & 0 & \sum z^2 &0 \\
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| 0 & \sum z^2 & 0 & \sum z^4 \\
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| \sum z^2 & 0 & \sum z^4 &0 \\
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| 0 &\sum z^4 & 0 & \sum z^6 \\
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| \end{pmatrix}=
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| \begin{pmatrix} 5 & 0 & 10 & 0 \\ 0 & 10 & 0 & 34 \\ 10 & 0 & 34 & 0 \\ 0 & 34 & 0 & 130\\ \end{pmatrix}
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| </math>
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| Now, the normal equations can be factored into two separate sets of equations, by rearranging rows and columns, with
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| :<math>
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| \mathbf {J^TJ}_{even}=\begin{pmatrix}5 & 10 \\ 10 & 34 \\\end{pmatrix} \quad \mathrm{and} \quad
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| \mathbf {J^TJ}_{odd}=\begin{pmatrix}10 & 34 \\ 34 & 130 \\\end{pmatrix}
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| </math>
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| :<math>
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| (\mathbf{J^TJ})^{-1}_\text{even}={1\over70}\begin{pmatrix}34 & -10\\ -10 & 5 \\\end{pmatrix}
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| \quad \mathrm{and} \quad | |
| (\mathbf{J^TJ})^{-1}_\text{odd}={1\over144}\begin{pmatrix}130 & -34\\ -34 & 10 \\\end{pmatrix}
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| </math>
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| The normal equations become
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| :<math>
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| \begin{pmatrix} {a_0 } \\ {a_2 } \\\end{pmatrix}_j=
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| {1\over70}\begin{pmatrix} 34 & -10\\ -10 & 5 \end{pmatrix}
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| \begin{pmatrix} 1&1&1&1&1\\4&1&0&1&4\\\end{pmatrix}
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| \begin{pmatrix}y_{j-2} \\ y_{j-1} \\ y_j \\ y_{j+1} \\ y_{j+2} \end{pmatrix}
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| </math> | |
| and
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| :<math>
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| \begin{pmatrix} a_1 \\ a_3 \\\end{pmatrix}_j=
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| {1\over144}\begin{pmatrix} 130 & -34\\ -34 & 10 \\\end{pmatrix}
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| \begin{pmatrix} -2&-1&0&1&2\\-8&-1&0&1&8\\\end{pmatrix}
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| \begin{pmatrix}y_{j-2} \\ y_{j-1} \\ y_j \\ y_{j+1} \\ y_{j+2} \\\end{pmatrix}
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| </math>
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| Multiplying out and removing common factors,
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| :<math>a_{0,j}={1\over35}(-3y_{j-2}+12y_{j-1}+17y_j+12y_{j+1}-3y_{j+2}) </math>
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| :<math>a_{1,j}={1\over12}(y_{j-2} - 8y_{j-1} +8y_{j+1} -y_{j+2})</math>
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| :<math>a_{2,j}={1\over14}(2y_{j-2} - y_{j-1} -2y_j -y_{j+1} +2y_{j+2})</math>
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| :<math>a_{3,j}={1\over12}(-y_{j-2} +2y_{j-1} -2y_{j+1} +y_{j+2})</math>
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| The coefficients of ''y'' in these expressions are known as [[convolution]] coefficients. They are elements of the matrix
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| :<math> \mathbf{C=(J^TJ)^{-1}J^T}</math>
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| In general,
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| :<math>(C \otimes y)_j\ = Y_j= \sum _{i=-(m-1)/2} ^{i=(m-1)/2}C_i\, y_{j+i}\qquad \frac{m+1}{2} \le j \le n-\frac{m-1}{2}</math>
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| Tables of convolution coefficients, calculated in the same way for ''m'' up to 25, were published for the [[Savitzky–Golay smoothing filter|Savitzky–Golay filter]] in 1964,<ref name="SG"/><ref name=Steinier/> The value of the central point, ''z'' = 0, is obtained from a single set of coefficients, ''a''<sub>0</sub> for smoothing, ''a''<sub>1</sub> for 1st. derivative etc. The numerical derivatives are obtained by differentiating Y. This means that the ''derivatives are calculated for the smoothed data curve''. For a cubic polynomial
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| :<math>Y = a_0 + a_1 z + a_2 z^2 + a_3 z^3= a_0 \text{ at } z = 0, x=\bar x</math>
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| :<math>\frac{{dY}}{{dx}} = \frac{1}{h}\left( {a_1 + 2a_2 z + 3a_3 z^3 } \right) = \frac{1}{h}a_1 \text{ at } z = 0, x=\bar x</math>
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| :<math>\frac{{d^2 Y}}{{dx^2 }} = \frac{1}{{h^2 }}\left( {2a_2 + 6a_3 z} \right) = \frac{2}{h^2}a_2 {\text{ at }}z =0, x=\bar x </math>
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| :<math>\frac{{d^3 Y}}{{dx^3 }} = \frac{6}{{h^3 }}a_3 </math>
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| In general, polynomials of degree (0 and 1),<ref group=note>Smoothing using the moving average is equivalent, with equally spaced points, to local fitting with a (sloping) straight line</ref> (2 and 3), (4 and 5) etc. give the same coefficients for smoothing and even derivatives. Polynomials of degree (1 and 2), (3 and 4) etc. give the same coefficients for odd derivatives.
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| Note. It is implicit in the above treatment that the data points are given equal weight. Technically, the [[objective function]]
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| :<math>U = \sum_i w_i(Y_i-y_i)^2 </math>
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| being minimized has unit weights, ''w''<sub>i</sub>=1. When weights are not all the same the normal equations become
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| :<math>\mathbf a = \left( \mathbf {J^T W} \mathbf{J} \right)^{-1} \mathbf{J^T W} \mathbf{y}\qquad W_{i,i} \ne 1</math>,
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| These equations do not have an analytical solution. To use non-unit weights, the coefficients of the fitting polynomial must be calculated individually for each data sub-set, using [[local regression]].
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| === Algebraic expressions ===
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| It is not necessary always to use the Savitzky–Golay tables. The summations in the matrix '''J'''<sup>T</sup>'''J''' can be evaluated in [[Faulhaber's formula|closed form]],
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| :<math>\sum_{-(m-1)/2}^{(m-1)/2} z^2= {m(m^2-1) \over 12}</math>
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| :<math>\sum z^4 = {m(m^2-1)(3m^2-7) \over 240} </math>
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| :<math>\sum z^6= {m(m^2-1)(3m^4-18m^2+31) \over 1344} </math>
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| so that algebraic formulae can be derived for the convolution coefficients.<ref>{{cite journal| last=Madden| first=Hannibal H.| title=Comments on the Savitzky–Golay convolution method for least-squares-fit smoothing and differentiation of digital data|journal=Anal. Chem.|year=1978|volume=50|issue=9|pages=1383–1386|doi=10.1021/ac50031a048|url=http://nsm1.nsm.iup.edu/jford/courses/CHEM421/Resources/CommentsOnTheSavitzkyGolayConvolutionMethodForLeastSquaresFitSmoothingAndDifferentiationOfDigitalData.pdf}}</ref><ref group=note>The expressions given here are different from those of Madden, which are given in terms of the variable m' = (m − 1)/2.</ref> For a cubic polynomial the expressions are (with (1 − ''m'')/2 ≤ ''i'' ≤ (''m'' − 1)/2)
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| :Smoothing: <math>
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| C_{0i} = \frac{
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| {\left( {3m^2 - 7 - 20i^2 } \right)/4}}
| |
| {{m\left( {m^2 - 4} \right)/3}}
| |
| </math>
| |
| :1st derivative: <math>
| |
| C_{1i} = \frac{
| |
| {5\left( {3m^4 - 18m^2 + 31} \right)i - 28\left( {3m^2 - 7} \right)i^3 }}
| |
| {{m\left( {m^2 - 1} \right)\left( {3m^4 - 39m^2 + 108} \right)/15}}
| |
| </math>
| |
| :2nd. derivative: <math>
| |
| C_{2i} = \frac{{12mi^2 - m\left( {m^2 - 1} \right)}}
| |
| {{m^2\left({m^2 - 1} \right) \left( {m^2 - 4} \right)/15}}
| |
| </math>
| |
| :3rd. derivative: <math>
| |
| C_{3i} = \frac{{ - \left( {3m^2 - 7} \right)i + 20i^3 }}
| |
| {{m\left( {m^2 - 1} \right)\left( {3m^4 - 39m^2 + 108} \right)/420}}.</math>
| |
| | |
| These functions are suitable for use with any curve that has an [[inflection point]]. Higher derivatives can be obtained. For example, a fourth derivative can be obtained by performing two passes of a second derivative function.<ref>Gans, pp. 153–157, "Repeated smoothing and differentiation"</ref>
| |
| | |
| === Use of orthogonal polynomials ===
| |
| An alternative to fitting ''m'' data points by a simple polynomial in the subsidiary variable, ''z'', is to use [[orthogonal polynomial]]s.
| |
| :<math>Y = b_0 P^0(z) + b_1 P^1(z) \cdots + b_k P^k(z).</math>
| |
| where ''P''<sup>0</sup> .. ''P''<sup>k</sup> is a set of mutually orthogonal polynomials of degree 0 .. ''k''. Full details on how to obtain expressions for the orthogonal polynomials and the relationship between the coefficients '''''b''''' and '''''a''''' are given by Guest.<ref name=Guest/> Expressions for the convolution coefficients are easily obtained because the normal equations matrix, '''''J'''''<sup>T</sup>'''''J''''', is a [[diagonal matrix]] as the product of any two orthogonal polynomials is zero by virtue of their mutual orthogonality. Therefore, each non-zero element of its inverse is simply the reciprocal the corresponding element in the normal equation matrix. The calculation is further simplified by using [[recursion]] to build orthogonal [[Gram polynomial]]s. The whole calculation can be coded in a few lines of [[Pascal (programming language)|PASCAL]], a computer language well-adapted for calculations involving recursion.<ref name=Gorry/>
| |
| | |
| === Treatment of first and last points ===
| |
| Savitzky–Golay filters are most commonly used to obtain the smoothed or derivative value at the central point, ''z'' = 0, using a single set of convolution coefficients. (''m'' − 1)/2 points at the start and end of the series cannot be calculated using this process. Various strategies can be employed to avoid this inconvenience.
| |
| * The data could be artificially extended by adding, in reverse order, copies of the first (''m'' − 1)/2 points at the beginning and copies of the last (''m'' − 1)/2 points at the end. For instance, with ''m'' = 5, two points are added at the start and end of the data ''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>.
| |
| :<span style="color:red;">''y''<sub>3</sub>,''y''<sub>2</sub></span>,''y''<sub>1</sub>, ... ,''y''<sub>''n''</sub>, <span style="color:red;">''y''<sub>''n''−1</sub>, ''y''<sub>''n''−2</sub></span>.
| |
| *Looking again at the fitting polynomial, it is obvious that data can be calculated for all values of ''z'' by using all sets of convolution coefficients for a single polynomial, a<sub>0</sub> .. a<sub>k</sub>.
| |
| :For a cubic polynomial
| |
| :<math>Y=a_0+a_1z + a_2 z^2 + a_3 z^3</math>
| |
| :<math>\frac{{dY}}{{dx}} = \frac{1}{h}\left( {a_1 + 2a_2 z + 3a_3 z^2 } \right) </math>
| |
| :<math>\frac{{d^2 Y}}{{dx^2 }} = \frac{1}{{h^2 }}\left( {2a_2 + 6a_3 z} \right) </math>
| |
| :<math>\frac{{d^3 Y}}{{dx^3 }} = \frac{6}{{h^3 }}a_3 {\text{ }}</math>
| |
| *Convolution coefficients for the missing first and last points can also be easily obtained.<ref name = Gorry>{{cite journal |first1=Gorry |last1=A. |year=1990 |title=General least-squares smoothing and differentiation by the convolution (Savitzky–Golay) method |journal=[[Analytical Chemistry (journal)|Analytical Chemistry]] |volume=62 |issue=6 |pages=570–573 |doi=10.1021/ac00205a007|url=http://nsm1.nsm.iup.edu/jford/courses/CHEM421/Resources/GeneralLeastSquaresSmoothingAndDifferentiationByTheConvolutionMethod.pdf}}</ref> This is also equivalent to fitting the first (''m''+1)/2 points with the same polynomial, and similarly for the last points.
| |
| | |
| == Two-dimensional convolution coefficients ==
| |
| Two-dimensional smoothing and differentiation can also be applied to tables of data values, such as intensity values in a photographic image which is composed of a rectangular grid of pixels.<ref name = DJT>{{cite web |url=http://www.doc.ic.ac.uk/research/technicalreports/2006/DTR06-8.pdf |title=Anisotropic Multidimensional Savitzky Golay kernels for Smoothing, Differentiation and Reconstruction |last1=Thornley |first1=David J |publisher=Imperial College Department of Computing, Technical report 2066/8}}</ref>
| |
| <ref>{{cite journal |last1=Ratzlaff |first1=Kenneth L. |last2=Johnson |first2=Jean T.|year=1989 |title=Computation of two-dimensional polynomial least-squares convolution smoothing integers |journal=Anal. Chem. |volume=61 |issue=11 |pages=1303–1305}}</ref> The trick is to transform part of the table into a row by a simple ordering of the indices of the pixels. Whereas the one-dimensional filter coefficients are found by fitting a polynomial in the subsidiary variable, ''z'' to a set of ''m'' data points. the two-dimensional coefficient are found by fitting a polynomial in subsidiary variables ''v'' and ''w'' to a set of ''m'' × ''m'' data points. The following example, for a bicubic polynomial and ''m'' = 5, illustrates the process, which parallels the process for the one dimensional case, above.<ref>{{cite web |url=http://research.microsoft.com/en-us/um/people/jckrumm/SavGol/SavGol.htm |title=Savitzky–Golay filters for 2D Images |last1=Krumm |first1=John |publisher=Microsoft Research, Redmond}}</ref>
| |
| :<math>v = \frac{x - \bar x}{h(x)}; w = \frac{y - \bar y}{h(y)}</math>
| |
| :<math>Y = a_{00} + a_{10}v + a_{01}w + a_{20}v^2 + a_{11}vw+a_{02}w^2 + a_{30}v^3 + a_{21}v^2w + a_{12}vw^2 + a_{03}w^3</math>
| |
| The square of 25 data values, ''d''<sub>1</sub> − '"d''<sub>25</sub>
| |
| :{| border="1" style="border-collapse:collapse;"
| |
| |-
| |
| | v||−2||−1||0||1||2
| |
| |-
| |
| |w
| |
| |-
| |
| |−2||''d''<sub>1</sub>||''d''<sub>2</sub>||''d''<sub>3</sub>||''d''<sub>4</sub>||''d''<sub>5</sub>
| |
| |-
| |
| |−1||''d''<sub>6</sub>||''d''<sub>7</sub>||''d''<sub>8</sub>||''d''<sub>9</sub>||''d''<sub>10</sub>
| |
| |-
| |
| |0||''d''<sub>11</sub>||''d''<sub>12</sub>||''d''<sub>13</sub>||''d''<sub>14</sub>||''d''<sub>15</sub>
| |
| |-
| |
| |1||''d''<sub>16</sub>||''d''<sub>17</sub>||''d''<sub>18</sub>||''d''<sub>19</sub>||''d''<sub>20</sub>
| |
| |-
| |
| |2||''d''<sub>21</sub>||''d''<sub>22</sub>||''d''<sub>23</sub>||''d''<sub>24</sub>||''d''<sub>25</sub>
| |
| |}
| |
| becomes a vector when the rows are placed one after another.
| |
| :'''d''' = (''d''<sub>1</sub> ... ''d''<sub>25</sub>)<sup>T</sup>
| |
| The Jacobian has 10 columns, one for each of the parameters ''a''<sub>00</sub> − ''a''<sub>03</sub> and 25 rows, one for each pair of ''v'' and ''w'' values. Each row has the form
| |
| :<math>J_\text{row} = 1\ v\ w\ v^2\ vw\ w^2\ v^3\ v^2w\ vw^2\ w^3</math>
| |
| The convolution coefficients are calculated as
| |
| :<math>\mathbf{C} = \left( \mathbf{J}^T \mathbf{J} \right)^{-1} \mathbf{J}^T</math>
| |
| The first row of '''C''' contains 25 convolution coefficients which can be multiplied with the 25 data values to provide a smoothed value for the central data point (13) of the 25.
| |
| | |
| A [[Matlab]]<ref>{{cite web |url=http://research.microsoft.com/en-us/um/people/jckrumm/SavGol/SavGolMatLab.m.htm |title=Compute Savitzky−Golay coefficients
| |
| |last1=Krumm |first1=John |publisher=Microsoft Research, Redmond}}</ref> routine for computing the coefficients is available. 3-dimensional filters can be obtained with a similar procedure.<ref name=DJT/>
| |
| | |
| == Some properties of convolution ==
| |
| #The sum of convolution coefficients for smoothing is equal to one. The sum of coefficients for odd derivatives is zero.<ref name =a7>Gans, Appendix 7</ref>
| |
| #The sum of squared convolution coefficients for smoothing is equal to the value of the central coefficient.<ref name =Ziegler/>
| |
| #Smoothing of a function leaves the area under the function unchanged.<ref name="a7" />
| |
| #Convolution of a symmetric function with even-derivative coefficients conserves the centre of symmetry.<ref name="a7" />
| |
| #Properties of derivative filters.<ref>{{cite journal|last=Luo|first=Jianwen|coauthors=Ying,Kui ; He, Ping ; Bai, Jing|title=Properties of Savitzky–Golay digital differentiators|journal=Digital Signal Processing|year=2005|volume=15|pages=122–136|doi=10.1016/j.dsp.2004.09.008|url=http://www.cecs.wright.edu/~phe/Research/DigitalSignalProcessing-05.pdf}}</ref>
| |
| | |
| === Signal distortion and noise reduction ===
| |
| It is inevitable that the signal will be distorted in the convolution process. From property 3 above, when data which has a peak is smoothed the peak height will be reduced and the [[Full width at half maximum|half-width]] will be increased. Both the extent of the distortion and S/N ([[signal-to-noise ratio]]) improvement:
| |
| *decrease as the degree of the polynomial increases
| |
| *increase as the width, ''m'' of the convolution function increases
| |
| | |
| For example, If the noise in all data points is uncorrelated and has a constant [[standard deviation]], σ, the standard deviation on the noise will be decreased by convolution with an ''m''-point smoothing function to<ref name=Ziegler>{{cite journal |last1=Ziegler |first1=Horst|year=1981 |title=Properties of Digital Smoothing Polynomial (DISPO) Filters |journal=Applied Spectroscopy |volume=35 |issue=1 |pages=88–92 }}</ref>
| |
| :polynomial degree 0 or 1:<math>\sqrt{{1\over m}} \sigma</math> (moving average)
| |
| :polynomial degree 2 or 3: <math>\sqrt{\frac{3(3m^2-7)}{4m(m^2-4)}} \sigma</math>.
| |
| Thus, with a 9-point linear function (moving average) two thirds of the noise is removed and with a 9-point quadratic/cubic smoothing function only about half the noise is removed. To remove two thirds of the noise with a quadratic/cubic smoothing function 21 points will be needed.
| |
| | |
| Although the moving average (polynomial order 0 or 1) gives the best noise reduction it is unsuitable for smoothing data which has curvature over ''m'' points. A quadratic filter is unsuitable for getting a derivative of a data curve with an [[inflection point]] because a quadratic polynomial does not have one. The optimal choice of polynomial order and number of convolution coefficients will be a compromise between noise reduction and distortion.<ref>{{cite journal |last1=Gans |first1=Peter |last2=Gill |first2=J. Bernard |year=1983 |title=Examination of the Convolution Method for Numerical Smoothing and Differentiation of Spectroscopic Data in Theory and in Practice |journal=Applied Spectroscopy |volume=37 |issue=6 |pages=515–520}}</ref>
| |
| | |
| ==== Multipass filters ====
| |
| One way to mitigate distortion and improve noise removal is to use a filter of smaller width and perform more than one convolution with it. For two passes of the same filter this is equivalent to one pass of a filter obtained by convolution of the original filter with itself.<ref>Gans, p. 153.</ref> The disadvantage of this process is that the equivalent filter width for ''n'' passes of an ''m''-point function is ''n''(''m'' − 1) + 1 so multipassing is subject to greater end-effects. Nevertheless, multipassing has been used to great advantage. For instance, some 40–80 passes on data with a signal-to-noise ratio of only 5 gave useful results.<ref>{{cite journal |last1=Procter |first1=Andrew|last2=Sherwood |first1=Peter M.A.|year=1980 |title=Smoothing of digital x-ray photoelectron spectra by an extended sliding least-squares approach |journal=Anal. Chem.|volume=52 |issue=14 |pages=2315–2321|DOI=10.1021/ac50064a018}}</ref> The noise reduction formulae given above do not apply because [[#Convolution and correlation|correlation]] between calculated data points increases with each pass.
| |
| | |
| === Frequency characteristics of convolution filters ===
| |
| [[File:FT 9 point cubic convolution function.png|thumb|160 px|Fourier transform of the 9-point quadratic/cubic smoothing function]]
| |
| Convolution maps to multiplication in the [[Fourier transform|Fourier]] [[Codomain|co-domain]]. The [[discrete Fourier transform]] of a convolution filter is a [[real-valued function]] which can be represented as
| |
| :<math>FT(\theta)= \sum_{j=(1-m)/2}^{j=(m-1)/2} C_j \cos(j\theta)</math>
| |
| θ runs from 0 to 180 [[Degree (angle)|degree]]s, after which the function merely repeats itself. The plot for a 9-point quadratic/cubic smoothing function is typical. At very low angle, the plot is almost flat, meaning that low-frequency components of the data will be virtually unchanged by the smoothing operation. As the angle increases the value decreases so that higher frequency components are more and more attenuated. This shows that the convolution filter can be described as a [[low-pass filter]]: the noise that is removed is primarily high-frequency noise and low-frequency noise passes through the filter.<ref>Gans, p. 207</ref> Some high-frequency noise components are attenuated more than others, as shown by undulations in the Fourier transform at large angles. This can give rise to small oscillations in the smoothed data.<ref name=Bromba>{{cite journal|last=Bromba|first=Manfred U.A|coauthors=Ziegler, Horst|title=Application hints for Savitzky–Golay digital smoothing filters|journal=Anal. Chem.|year=1981|volume=53|issue=11|pages=1583–1586|doi=10.1021/ac00234a011}}</ref>
| |
| | |
| === Convolution and correlation ===
| |
| Convolution affects the correlation between errors in the data. The effect of convolution can be expressed as a linear transformation.
| |
| :<math>Y_j\ = \sum_{i=-(m-1)/2}^{i=(m-1)/2} C_i\, y_{j+i}</math>
| |
| By the law of [[error propagation]], the [[variance-covariance matrix]] of the data, '''A''' will be transformed into '''B''' according to
| |
| :<math>\mathbf B=\mathbf C \mathbf A\mathbf C^T</math>
| |
| To see how this applies in practice, consider the effect of a 5-point quadratic smoothing function for the first five calculated points, ''Y''<sub>3</sub> − ''Y''<sub>7</sub>.
| |
| :<math>
| |
| \mathbf C ={1 \over 35}
| |
| \begin{pmatrix}
| |
| - 3 & 12 & 17 & 12 & - 3 & 0 &0&0&0&\dots \\
| |
| 0 & - 3 & 12 & 17 & 12 & - 3 &0&0&0&\dots \\
| |
| 0 &0& - 3 & 12 & 17 & 12 & - 3&0&0&\dots\\
| |
| 0 &0&0& - 3 & 12 & 17 & 12 & - 3&0&\dots\\
| |
| 0&0 &0&0& - 3 & 12 & 17 & 12 & - 3&\dots\\
| |
| \dots
| |
| \end{pmatrix}
| |
| </math> | |
| If one assumes that the data points have equal [[variance]] and that there is no correlation between them, '''A''' will be an [[identity matrix]] multiplied by a constant, σ<sup>2</sup>, the variance at each point. In that case, <math>\mathbf B=\sigma^2 \mathbf C \mathbf C^T</math>. The [[Correlation and dependence|correlation coefficient]]s, <math>\rho_{ij}=B_{ij}/\sigma^2 (i\ne j)</math>, between calculated points ''i'' and ''j'' will be obtained by [[vector multiplication]] of rows ''i'' and ''j'' of '''C'''. | |
| | |
| :<math>
| |
| \rho_{ij} =
| |
| \begin{pmatrix}
| |
| i\\
| |
| 2&0.27\\
| |
| 3&0.03&0.27\\
| |
| 4&-0.06&0.03&0.27\\
| |
| 5&0.01&-0.06&0.03&0.27\\
| |
| j&1&2&3&4\\
| |
| \end{pmatrix}
| |
| </math>
| |
| In this example the correlation coefficient between adjacent points is 0.27. Thus, the calculated values are correlated even when the observed values are not correlated. The same pattern applies to the rest of the calculated points. The correlation extends over ''m'' − 1 calculated points at a time.<ref>Gans, p.157</ref>
| |
| | |
| == See also ==
| |
| *[[Kernel smoother]] – Different terminology for many of the same processes, used in [[statistics]]
| |
| *[[Local regression]] — the LOESS and LOWESS methods
| |
| *[[Numerical differentiation]] – Application to differentiation of [[function (mathematics)|function]]s
| |
| *[[Smoothing spline]]
| |
| *[[Stencil (numerical analysis)]] – Application to the solution of differential equations
| |
| | |
| == Notes ==
| |
| {{reflist|group=note}}
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *{{cite book|url=http://books.google.com/books/about/Data_fitting_in_the_chemical_sciences.html?id=fRcvAQAAIAAJ|title=Data fitting in the chemical sciences: By the method of least squares|isbn=9780471934127|author1=Gans|first1=Peter|year=1992}}
| |
| | |
| ==External links==
| |
| * [http://www.statistics4u.info/fundstat_eng/cc_filter_savgolay.html Savitzky–Golay filter in Fundamentals of Statistics]
| |
| | |
| == Appendix ==
| |
| | |
| ===Tables of selected convolution coefficients===
| |
| | |
| Consider a set of data points (''x<sub>j</sub>'', ''y<sub>j</sub>'')<sub>1 ≤ ''j'' ≤ ''n''</sub>. The Savitzky–Golay tables refer to the case that the step ''x''<sub>''j''</sub> − ''x''<sub>''j'' − 1</sub> is constant, ''h''. Examples of the use of the so-called convolution coefficients, with a cubic polynomial and a window size, ''m'', of 5 points are as follows.
| |
| :Smoothing: <math>Y_j = \frac{1}{35} (-3 \times y_{j - 2} + 12 \times y_{j - 1} + 17 \times y_j + 12 \times y_{j + 1} -3 \times y_{j + 2})</math> ;
| |
| :1st derivative: <math>Y'_j = \frac{1}{12h} (1 \times y_{j - 2} - 8 \times y_{j - 1} + 0 \times y_j + 8 \times y_{j + 1} - 1 \times y_{j + 2})</math> ;
| |
| :2nd. derivative: <math>Y''_j = \frac{1}{7h^2} (2 \times y_{j - 2} - 1 \times y_{j - 1} - 2 \times y_j - 1 \times y_{j + 1} + 2 \times y_{j + 2})</math>.
| |
| Selected values of the convolution coefficients for polynomials of degree 1,2,3, 4 and 5 are given in the following tables.
| |
| | |
| {|
| |
| |
| |
| {| class="wikitable"
| |
| |+ Coefficients for smoothing
| |
| |-
| |
| ! scope="row" | Degree
| |
| ! colspan="3" scope="col" | 2/3 (quadratic/cubic)
| |
| ! colspan="3" scope="col" | 4/5 (quartic/quintic)
| |
| |-
| |
| ! scope="row" | Window size
| |
| ! scope="col" | 5 !! scope="col" | 7 !! scope="col" | 9
| |
| ! scope="col" | 7 !! scope="col" | 9
| |
| |-
| |
| ! scope="row" | −4
| |
| | || || −21 || || 15
| |
| |-
| |
| ! scope="row" | −3
| |
| | || −2 || 14 || 5 || −55
| |
| |-
| |
| ! scope="row" | −2
| |
| | −3 || 3 || 39 || −30 || 30
| |
| |-
| |
| ! scope="row" | −1
| |
| | 12 || 6 || 54 || 75 || 135
| |
| |-
| |
| ! scope="row" | 0
| |
| | 17 || 7 || 59 || 131 || 179
| |
| |-
| |
| ! scope="row" | 1
| |
| | 12 || 6 || 54 || 75 || 135
| |
| |-
| |
| ! scope="row" | 2
| |
| | −3 || 3 || 39 || −30 || 30
| |
| |-
| |
| ! scope="row" | 3
| |
| | || −2 || 14 || 5 || −55
| |
| |-
| |
| ! scope="row" | 4
| |
| | || || −21 || || 15
| |
| |-
| |
| ! scope="row" | Normalisation
| |
| | 35 || 21 || 231 || 231 || 429
| |
| |}
| |
| | |
| |
| |
| {| class="wikitable"
| |
| |+ Coefficients for 1<sup>st</sup> derivative
| |
| |-
| |
| ! scope="row" | Degree
| |
| ! colspan="3" scope="col" | 1/2 (linear/quadratic)
| |
| ! colspan="3" scope="col" | 3/4 (cubic/quartic)
| |
| |-
| |
| ! scope="row" | Window size
| |
| ! scope="col" | 5 !! scope="col" | 7 !! scope="col" | 9
| |
| ! scope="col" | 5 !! scope="col" | 7 !! scope="col" | 9
| |
| |-
| |
| ! scope="row" | −4
| |
| | || || −4 || || || 86
| |
| |-
| |
| ! scope="row" | −3
| |
| | || −3 || −3 || || 22 || −142
| |
| |-
| |
| ! scope="row" | −2
| |
| | −2 || −2 || −2 || 1 || −67 || −193
| |
| |-
| |
| ! scope="row" | −1
| |
| | −1 || −1 || −1 || −8 || −58 || −126
| |
| |-
| |
| ! scope="row" | 0
| |
| | 0 || 0 || 0 || 0 || 0 || 0
| |
| |-
| |
| ! scope="row" | 1
| |
| | 1 || 1 || 1 || 8 || 58 || 126
| |
| |-
| |
| ! scope="row" | 2
| |
| | 2 || 2 || 2 || −1 || 67 || 193
| |
| |-
| |
| ! scope="row" | 3
| |
| | || 3 || 3 || || −22 || 142
| |
| |-
| |
| ! scope="row" | 4
| |
| | || || 4 || || || −86
| |
| |-
| |
| ! scope="row" | Normalisation
| |
| | 10 || 28 || 60 || 12 || 252 || {{formatnum:1188}}
| |
| |}
| |
| |}
| |
| | |
| {|
| |
| |
| |
| {| class="wikitable"
| |
| |+ Coefficients for 2<sup>nd</sup> derivative
| |
| |-
| |
| ! scope="row" | Degree
| |
| ! colspan="3" scope="col" | 2/3 (quadratic/cubic)
| |
| ! colspan="3" scope="col" | 4/5 (quartic/quintic)
| |
| |-
| |
| ! scope="row" | Window size
| |
| ! scope="col" | 5 !! scope="col" | 7 !! scope="col" | 9
| |
| ! scope="col" | 5 !! scope="col" | 7 !! scope="col" | 9
| |
| |-
| |
| ! scope="row" | −4
| |
| | || || 28 || || || {{formatnum:−4158}}
| |
| |-
| |
| ! scope="row" | −3
| |
| | || 5 || 7 || || −117 || {{formatnum:12243}}
| |
| |-
| |
| ! scope="row" | −2
| |
| | 2 || 0 || −8 || −3 || 603 || {{formatnum:4983}}
| |
| |-
| |
| ! scope="row" | −1
| |
| | −1 || −3 || −17 || 48 || −171 || {{formatnum:−6963}}
| |
| |-
| |
| ! scope="row" | 0
| |
| | −2 || −4 || −20 || −90 || −630 || {{formatnum:−12210}}
| |
| |-
| |
| ! scope="row" | 1
| |
| | −1 || −3 || −17 || 48 || −171 || {{formatnum:−6963}}
| |
| |-
| |
| ! scope="row" | 2
| |
| | 2 || 0 || −8 || −3 || 603 || {{formatnum:4983}}
| |
| |-
| |
| ! scope="row" | 3
| |
| | || 5 || 7 || || −117 || {{formatnum:12243}}
| |
| |-
| |
| ! scope="row" | 4
| |
| | || || 28 || || || {{formatnum:−4158}}
| |
| |-
| |
| ! scope="row" | Normalisation
| |
| | 7 || 42 || 462 || 36 || 99|| 4719
| |
| |}
| |
| |
| |
| {| class="wikitable"
| |
| |+ Coefficients for 3<sup>rd</sup> derivative
| |
| |-
| |
| ! Degree
| |
| ! colspan="3" | 3/4 (cubic/quartic)!! colspan="3" | 5/5 (quintic/sextic)
| |
| |-
| |
| !Window size!! 5 !! 7 !! 9!! 7 !! | 9
| |
| |-
| |
| ! −4
| |
| | || || −14 || || 100
| |
| |-
| |
| !−3
| |
| | || −1 || 7 || 1 || −457
| |
| | |
| |-
| |
| !−2
| |
| | −1 || 1 || 13 || −8 || 256
| |
| |-
| |
| !−1
| |
| | 2 || 1 || 9 || 13 || 459
| |
| |-
| |
| !0
| |
| | 0 || 0 || 0 || 0 || 0
| |
| |-
| |
| !1
| |
| | −2 || −1 || −9 || −13 || −459
| |
| |-
| |
| !2
| |
| | 1 || −1 || −13 ||8 || −256
| |
| |-
| |
| !3
| |
| | || 1 || −7 ||−1 || 457
| |
| |-
| |
| !4
| |
| | || || 14 || || −100
| |
| |-
| |
| ! Normalisation
| |
| | 2 || 6 || 198 || 8 ||1144
| |
| |}
| |
| |
| |
| {| class="wikitable"
| |
| |+ Coefficients for 4<sup>th</sup> derivative
| |
| |-
| |
| ! Degree
| |
| ! colspan="2" | 4/5 (quartic/quintic)
| |
| |-
| |
| !Window size!! 7 !! 9
| |
| |-
| |
| ! −4
| |
| | ||14
| |
| |-
| |
| !−3
| |
| |3 || −21
| |
| |-
| |
| !−2
| |
| | −7 || −11
| |
| |-
| |
| !−1
| |
| | 1 || 9
| |
| |-
| |
| !0
| |
| | 6 || 18
| |
| |-
| |
| !1
| |
| | 1 || 9
| |
| |-
| |
| !2
| |
| | -7 || −11
| |
| |-
| |
| !3
| |
| |3 || −21
| |
| |-
| |
| !4
| |
| | ||14
| |
| |-
| |
| ! Normalisation
| |
| | 11 || 143
| |
| |}
| |
| |}
| |
| | |
| {{DEFAULTSORT:Savitzky-Golay filter for smoothing and differentiation}}
| |
| [[Category:Filter theory]]
| |
| [[Category:Signal processing]]
| |
| [[Category:Regression analysis]]
| |