Lsq2top: Difference between revisions
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===Purpose=== | ===Purpose=== | ||
Fits a polynomial to the top/(bottom) of data. | Fits a polynomial to the top/(bottom) of data. | ||
===Synopsis=== | ===Synopsis=== | ||
:[b,resnorm,residual,options] = lsq2top(x,y,order,res,options) | :[b,resnorm,residual,options] = lsq2top(x,y,order,res,options) | ||
===Description=== | ===Description=== | ||
LSQ2TOP is an iterative least squares fitting algorithm. It is based on a weighted least squares approach where the weights are determined at each step. At initialization the weights are all 1, then a polynomial is fit through the data cloud using least squares. When fitting to the top of a data cloud, data points with a residual significantly below a defined limit (i.e. the points below the polynomial fit line) are given a small weighting. Therefore, on subsequent iterations these data points are weighted less in the fit, and the fit line moves to fit to the top of the data cloud. | LSQ2TOP is an iterative least squares fitting algorithm. It is based on a weighted least squares approach where the weights are determined at each step. At initialization the weights are all 1, then a polynomial is fit through the data cloud using least squares. When fitting to the top of a data cloud, data points with a residual significantly below a defined limit (i.e. the points below the polynomial fit line) are given a small weighting. Therefore, on subsequent iterations these data points are weighted less in the fit, and the fit line moves to fit to the top of the data cloud. | ||
Input x is the independent variable e.g. a ''M''x''1'' vector corresponding to a frequency or wavelength axis. Input y is the dependent variable e.g. a ''M''x''1'' vector corresponding to a measured spectrum. Input order is a scalar defining the order of polynomial to be fit e.g. y = P(x), and res is a scalar approximation of the fit residual e.g. noise level. Input options is discussed below. Note that the function can be used to fit to the top or bottom of a data cloud by changing trbflag in options. | Input x is the independent variable e.g. a ''M''x''1'' vector corresponding to a frequency or wavelength axis. Input y is the dependent variable e.g. a ''M''x''1'' vector corresponding to a measured spectrum. Input order is a scalar defining the order of polynomial to be fit e.g. y = P(x), and res is a scalar approximation of the fit residual e.g. noise level. Input options is discussed below. Note that the function can be used to fit to the top or bottom of a data cloud by changing trbflag in options. | ||
The outputs are b, the regression coefficients [highest order term corresponds to b(1) and the intercept corresponds to b(end)], resnorm is the squared 2-norm of the residual, and residual is the fit residuals = y - P(x). The options ouput is the input options echoed back, the field initwt may have been modified. | The outputs are b, the regression coefficients [highest order term corresponds to b(1) and the intercept corresponds to b(end)], resnorm is the squared 2-norm of the residual, and residual is the fit residuals = y - P(x). The options ouput is the input options echoed back, the field initwt may have been modified. | ||
===Options=== | ===Options=== | ||
* '''display''': [ 'off' | {'on'} ] governs level of display to command window. | * '''display''': [ 'off' | {'on'} ] governs level of display to command window. | ||
* '''trbflag''': [ 'top' | {'bottom'} ] top or bottom flag, tells algorithm to fit the polynomials, y = P(x), to the top or bottom of the data cloud. | * '''trbflag''': [ 'top' | {'bottom'} ] top or bottom flag, tells algorithm to fit the polynomials, y = P(x), to the top or bottom of the data cloud. | ||
* '''tsqlim''': [ 0.99 ] limit that governs whether a data point is significantly outside the fit residual defined by input res. | * '''tsqlim''': [ 0.99 ] limit that governs whether a data point is significantly outside the fit residual defined by input res. | ||
* '''stopcrit''': [1e-4 1e-4 1000 360] stopping criteria, iteration is continued until one of the stopping criterion is met: [(relative tolerance) (absolute tolerance) (maximum number of iterations) (maximum time [seconds])]. | * '''stopcrit''': [1e-4 1e-4 1000 360] stopping criteria, iteration is continued until one of the stopping criterion is met: [(relative tolerance) (absolute tolerance) (maximum number of iterations) (maximum time [seconds])]. | ||
* '''initwt''': [ ] empty or Mx1 vector of initial weights (0<=w<=1). | * '''initwt''': [ ] empty or Mx1 vector of initial weights (0<=w<=1). | ||
===Algorithm=== | ===Algorithm=== | ||
For order = 1 and fitting to the top of a data cloud, LSQ2TOP finds the vector | |||
The weighting is determined by first estimating the residuals for each data point ''j'' as | For order = 1 and fitting to the top of a data cloud, LSQ2TOP finds the vector <math>\mathbf{b}=\left[ \begin{matrix} | ||
b_{1} & b_{2} \\ | |||
where | \end{matrix} \right]</math> | ||
that minimizes <math>\left( \mathbf{y}-\mathbf{x}b_{1}-\mathbf{1}b_{2} \right)^{T}\mathbf{W}\left( \mathbf{y}-\mathbf{x}b_{1}-\mathbf{1}b_{2} \right)</math> | |||
where '''W''' is a diagonal weighting matrix whose elements are initially 1 and then are modified on each subsequent iteration. | |||
The weighting is determined by first estimating the residuals for each data point ''j'' as <math>residual_{j}=\mathbf{y}_{j}-\mathbf{x}_{j}b_{1}-b_{2}</math> | |||
and defining <math>t_{j}=residual_{j}/res</math> | |||
where ''res'' is the input res. A corresponding t-statistic from a t-table is estimated using the following | |||
<pre>tsqst = ttestp(1-options.tsqlim,5000,2);</pre> | |||
where <math>t_{table}</math> | |||
is tsqst. The elements of '''W''' are then given by <math>w_{j}={1}/{\left( 0.5+t_{j}/t_{table} \right)}\;</math> | |||
for data points with <math>t_{j}<t_{table}</math>, and is a 1 otherwise. Therefore, the weighting is smaller for points far below the fit line. | |||
The procedure can be modified to fit to the bottom of a data cloud by changing options.trbflag. | The procedure can be modified to fit to the bottom of a data cloud by changing options.trbflag. | ||
===See Also=== | ===See Also=== | ||
[[ | |||
[[baseline]], [[baselinew]], [[fastnnls]], [[fastnnls_sel]], [[med2top]] | |||
Latest revision as of 09:42, 21 September 2011
Purpose
Fits a polynomial to the top/(bottom) of data.
Synopsis
- [b,resnorm,residual,options] = lsq2top(x,y,order,res,options)
Description
LSQ2TOP is an iterative least squares fitting algorithm. It is based on a weighted least squares approach where the weights are determined at each step. At initialization the weights are all 1, then a polynomial is fit through the data cloud using least squares. When fitting to the top of a data cloud, data points with a residual significantly below a defined limit (i.e. the points below the polynomial fit line) are given a small weighting. Therefore, on subsequent iterations these data points are weighted less in the fit, and the fit line moves to fit to the top of the data cloud.
Input x is the independent variable e.g. a Mx1 vector corresponding to a frequency or wavelength axis. Input y is the dependent variable e.g. a Mx1 vector corresponding to a measured spectrum. Input order is a scalar defining the order of polynomial to be fit e.g. y = P(x), and res is a scalar approximation of the fit residual e.g. noise level. Input options is discussed below. Note that the function can be used to fit to the top or bottom of a data cloud by changing trbflag in options.
The outputs are b, the regression coefficients [highest order term corresponds to b(1) and the intercept corresponds to b(end)], resnorm is the squared 2-norm of the residual, and residual is the fit residuals = y - P(x). The options ouput is the input options echoed back, the field initwt may have been modified.
Options
- display: [ 'off' | {'on'} ] governs level of display to command window.
- trbflag: [ 'top' | {'bottom'} ] top or bottom flag, tells algorithm to fit the polynomials, y = P(x), to the top or bottom of the data cloud.
- tsqlim: [ 0.99 ] limit that governs whether a data point is significantly outside the fit residual defined by input res.
- stopcrit: [1e-4 1e-4 1000 360] stopping criteria, iteration is continued until one of the stopping criterion is met: [(relative tolerance) (absolute tolerance) (maximum number of iterations) (maximum time [seconds])].
- initwt: [ ] empty or Mx1 vector of initial weights (0<=w<=1).
Algorithm
For order = 1 and fitting to the top of a data cloud, LSQ2TOP finds the vector that minimizes where W is a diagonal weighting matrix whose elements are initially 1 and then are modified on each subsequent iteration.
![{\displaystyle \mathbf {b} =\left[{\begin{matrix}b_{1}&b_{2}\\\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b1552839940141f8a5430ffd4709fada1a5aef3)
The weighting is determined by first estimating the residuals for each data point j as and defining where res is the input res. A corresponding t-statistic from a t-table is estimated using the following
tsqst = ttestp(1-options.tsqlim,5000,2);
where is tsqst. The elements of W are then given by for data points with , and is a 1 otherwise. Therefore, the weighting is smaller for points far below the fit line.
The procedure can be modified to fit to the bottom of a data cloud by changing options.trbflag.