Curve Fitting in NCSS
Using NCSS as curve fitting software by using the several tools available for finding and modeling the best (often nonlinear) fit of a response (Y) to one or more independent variables (X’s). Each curve fitting procedure is easy-to-use and validated for accuracy. Use the links below to jump to a specific online curve fitting topic. To see how these tools can benefit you, we recommend you download and install the free trial of NCSS.
Curve fitting refers to finding an appropriate mathematical model that expresses the relationship between a dependent variable Y and a single independent variable X (or group of X’s) and estimating the values of its parameters using nonlinear regression.
This page provides a general overview of the capabilities of NCSS as curve fitting software for analysis. If you would like to examine the formulas and technical details relating to a specific NCSS procedure, click on the corresponding ‘[Documentation PDF]’ link under each heading to load the complete procedure documentation. There you will find formulas, references, discussions, and examples or tutorials describing the procedure in detail.
Curve Fitting – General
This procedure is a general purpose curve fitting procedure providing many new technologies that are not readily available in most other statistical packages. Our online curve fitting software is pre-programmed to fit over forty common mathematical models including growth models like linear-growth and Michaelis-Menten. It also fits many approximating models such as regular polynomials, piecewise polynomials and polynomial ratios. In addition to these pre-programmed models, it also fits your custom-written models.
The Curve Fitting – General procedure includes several innovative features. Our online curve fitting software can fit curves to several groups of data simultaneously. This procedure compares fitted models across groups using graphics and numerical tests such as an approximate F-test for curve coincidence, and a computer-intensive randomization test that compares curve coincidence and individual parameter values. Further, this routine computes bootstrap confidence intervals for parameter values, predicted means, and predicted values using the latest computer-intensive bootstrapping technology.
Setup Window for the Curve Fitting – General Procedure
Example Output for the Curve Fitting – General Procedure
A Curve Fitting Plot from the Curve Fitting – General Procedure
Ratio of Polynomials Search – One Variable
This procedure searches through hundreds of potential curves looking for the model that best fits your data. The procedure is heuristic in nature, but does well at selecting appropriate models.
A general class of models called the ratio of polynomials provides a wide variety of curves on which a search can be based. Normally, fitting these models is a slow, iterative process. However, using a heuristic shortcut, an approximate solution may be found very quickly so that a large number of models may be searched in a short period of time. After the best fitting model is found, you can use the Ratio of Polynomials Fit – One Variable procedure to provide a detailed analysis of the model.
For each model, various transformations of X and Y can be tried. This expands the number of models that may be tried to several hundred.
The general formula for the ratio of polynomials model fit is
Here g(Y) and f(X) represent power transformations of Y and X such as LOG(X), SQRT(X), etc. The parameters A0, A1, A2, …, B5 are constants that are estimated from the data. The value e represents the error or residual of the observations. By setting some constants to zero, various simplified models are obtained. For example, if only A0 and A1 are nonzero, the familiar linear model, Y=A0+A1X+e, is obtained.
Ratio of Polynomials Fit – One Variable
This program fits a model that is the ratio of two polynomials of up to fifth order. Examples of this type of model are:
These models approximate many different curves. They offer a much wider variety of curves than the usual polynomial models. Since these are approximating curves and have no physical interpretation, care must be taken outside the range of the data. You must study the resulting model graphically to determine that the model behaves properly between data points.
Usually you would use the Ratio of Polynomials Search – One Variable procedure first to find an appropriate model and then fit that model with this procedure.
Example Plots from the Ratio of Polynomials Fit – One Variable Procedure
Ratio of Polynomials Search – Many Variables
This procedure is similar to the Ratio of Polynomials Search – One Variable procedure, except that it is more general in that it models up to four independent variables. This procedure searches through hundreds of potential curves to find the best fit using a heuristic algorithm.
After the best fitting model is found, you can use the Ratio of Polynomials Fit – Many Variables procedure to further examine the model.
Ratio of Polynomials Fit – Many Variables
This procedure fits a model that is the ratio of two polynomials of up to fifth order. Instead of a single independent variable, these polynomials may involve up to four independent variables (U, V, W, and X). An example of this type of model is:
Usually you would use the Ratio of Polynomials Search – Many Variables procedure first to find an appropriate model and then fit and further analyze that model with this procedure.
Example Plots from the Ratio of Polynomials Fit – Many Variables Procedure
Sum of Functions Models
This procedure fits models that are the ratio of two linear expressions. The general form of a model is:
where f(X), g(Y), and h(X) are standard functions such as SIN(X), LN(X+1), SQRT(X/2), etc. The A0, A1, …, B5 are constants (parameters) to be estimated from the data.
These models approximate many different curves. They offer a much wider variety of curves than the usual polynomial models.
Example Plots from the Sum of Functions Models Procedure
A nonlinear regression procedure is also available in NCSS for nonlinear curve fitting. Click here to go to the Nonlinear Regression procedure description.
Function plots can be used to view a plot of a desired mathematical function of X. Some of the available functions are Sin, Cos, Exponent, Log, Tan, Square Root, and so on.
An Example Plot from the Function Plots Procedure
Scatter Plot Matrix for Curve Fitting
One of the first tasks in curve fitting online is to graphically inspect your data. This procedure allows you to view scatter plots of various transformations of both X and Y. Usually, your first choice would be to look for transformations of X and Y that yield a straight line. These plots are shown in matrix format.
The available transformations for both X and Y are the inverse, the inverse of the square, the inverse of the square root, the natural log, the square root, and the square.
Scatter Plot Matrix for Curve Fitting Example
Distance (cm) = -125.3911 + 492.0476*Time (sec) + 486.55399*(Time (sec)-0.51619)2
In this model, note how the quadratic term is written. for Time (sec) is written as (Time (sec) -0.51619)2. This means that the polynomial has been centered. The values of Time (sec) were “centered” by subtracting the mean.
Centering polynomials is a standard technique used when fitting linear models with higher-order terms. It leads to the same model predictions, but does a better job of estimating the model coefficients.
In this example, the residual analysis pointed to a problem, and fitting a polynomial model made sense. In most real-life scenarios, fitting the best possible model when there are unusual patterns in data is not as straightforward.
For example, you might need to apply a transformation to the response or the predictor. Or you might be missing other important effects that explain the relationship. The decision on how to proceed with the analysis should be guided by subject matter knowledge and the context of the problem.
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About ZunZun.comHere you can curve and surface fit your 2D and 3D data online with a rich set of error histograms, error plots, curve plots, surface plots, contour plots, VRML, and source code.
The author, James Phillips began to curve and surface fit when he was in R&D around 1992. These were mostly X-ray transmission and backscatter curve and surface data sets from the measurement of steel and aluminum thickness. He also did work with X-ray fluorescence of zinc coatings on steel. The software tools he had on hand at the time were too expensive for general use. Since he was learning to program computers, he started writing curve fitting software, first in C and then in C++. He left Tokyo, Japan in early 2000, returning to Birmingham, Alabama, and started programming in Python.
Linear vs. Non-linear Regression
"Linear regression" is used for equations that are linear *in the coefficients*. Here are some specific examples:
y = mx + b
y = aX^0 + bX^1 + cX^2 + dX^3
z = a + bX + cY + dXY
These equations are all of the general form:
result = coefficient_1 * function_1 + coefficient_2 * function_2 + coefficient_3 * function_3 + ...
where function_1 can be x^2, function_2 can be sin(x), etc., etc.
Here is an example equation that is NOT linear in the coefficients;
y = exp(a*x)
where we want to find the coefficient "a." The format of this equation cannot be used for linear regression, because it is not in the general form just discussed.
This is where "non-linear regression" comes into play. The basic idea is:
1) Make an initial guess for the value of "a"
2) Using our data set, calculate the sum of squared error (SSQ)
3) Try to change a so as to reduce the SSQ calculated in step 2 above (sometimes using derivatives)
4) Repeat steps 1 through 3 until the SSQ is as low as we can make it
Note that step 4 means repeated iteration, so non-linear regression is often referred to as "using iterative methods," which means it is computationally more expensive than linear regression and with large data sets can take a long time.
Multiple Linear Regression
Suppose you have 2-dimensional XY data, and want to fit a straight line to this data. The equation is commonly written as:
y = mx + b
This can be rewritten in polynomial form as
y = ax^0 + bx^1
A quadratic is then
y = ax^0 + bx^1 + cx^2
and a cubic is then
y = ax^0 + bx^1 + cx^2 + dx^3
and so on.
Straight lines are done, let's move on. For multiple linear regression, the key idea is the data formatting for the regression.
Let's say you want to fit your X and Y 2D data to this equation:
y = ax^3 + bsin(x)
Format your data as columns of
and again regress against Y.
Now for n-dimensional regression. Let's do a simple XYZ 3D fit first. Start with the equation
z = ax + by
Format your data as columns of
and regress against Z. All done.
For a more complex 3D problem, say
Z = a + bcos(X*Y) + cexp(X/Y)
format your data as columns of
1 cos(X*Y) exp(X/Y)
and regress against Z.
Absolute vs. Relative Error
Start with absolute error, since it is simple.
absolute error = predicted value - actual value
So if your curve fit predicts an output of 11.0 when the actual value is 10.0, then
absolute error = predicted - actual = 11.0 - 10.0 = 1.0
So in this case absolute error is 1.0.
Relative error is
absolute error / actual
So using the above example,
relative error = absolute error / actual = 1.0 / 10.0 = 0.1
Multiply by 100 to get percent error.
percent error = relative error * 100 = 0.1 * 100 = 10%
In this example, the curve fit is off by 10 percent.
Online Calculator Curve Fitting
Online calculator for curve fitting with least square methode for linear, polynomial, power, gaussian, exponential and fourier curves. Adaptation of the functions to any measurements.
Different functions can be adapted to data with the calculator: linear curve fit, polynomial curve fit, curve fit by Fourier series, curve fit by Gaussian distribution and power law curve fit. Through the selection of further fit functions the functions are added to the chart. The style of the functions and points can be selected by means of the style information.
The screenshot function allows to copy the diagram to an image. From the image it is possible to store or print the diagram.
Number of digits =
An alternative input is possible with load data from file. The values may be separated comma or space or semicolon. The values must be given pairwise x1,y1,x2,y2...
Linear Line curve fit
Power Law curve fit
Gauss Function curve fit
Polynomial curve fit
e-function curve fit
Square e-function curve fit
Fourier curve fit
Credentials - Imprint - Contact - HomeSours: https://elsenaju.eu/Calculator/online-curve-fit.htm
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Interactive Curve Fitting
To interactively fit a curve, follow the steps in this simple example:
Load some data at the MATLAB® command line.
Open the Curve Fitting app. Enter:
In the Curve Fitting app, select X Data and Y Data.
Curve Fitting app creates a default interpolation fit to the data.
Choose a different model type using the fit category drop-down list, e.g., select Polynomial.
Try different fit options for your chosen model type.
Select > .
Curve Fitting app creates a file in the Editor containing MATLAB code to recreate all fits and plots in your interactive session.
For more information about fitting curves in the Curve Fitting app, see Interactive Curve and Surface Fitting.
For details and examples of specific model types and fit analysis, see the following sections:
Programmatic Curve Fitting
To programmatically fit a curve, follow the steps in this simple example:
Load some data.
Create a fit using the function, specifying the variables and a model type (in this case is the model type).f = fit( temp, thermex, 'rat23' )
Plot your fit and the data.plot( f, temp, thermex ) f( 600 )
To learn what functions you can use to create and work with fits, see: Curve and Surface Fitting.
For details and examples of specific model types and fit analysis, see the following sections:
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