**Quickly Find the Best Equations that Describe Your Data**

TableCurve 2D® gives engineers and researchers the power to find the ideal model for even the most complex data, by putting thousands of equations at their fingertips.

TableCurve 2D’s built-in library includes a wide array of linear and nonlinear models for any application including equations that may never have been considered, from simple linear equations to high order Chebyshev polynomials.

**Learn More…**

## Overview

TableCurve 2D is the automatic choice for curve-fitting and data modeling for critical research.

TableCurve 2D’s state-of-the-art data fitting includes capabilities not found in other software packages:

- A 38-digit precision math emulator for properly fitting high order polynomials and rationals
- A robust fitting capability for nonlinear fitting that effectively copes with outliers and a wide dynamic Y data range
- An AI Expert option that automatically selects the appropriate peak, transition or kinetics models for you

**Automation Takes The Trial and Error Out of Curve Fitting**

Fit all of TableCurve 2D’s 3,665 built-in equations or just the ones you need in seconds! With TableCurve 2D, a single mouse click is all it takes to start the automated curve fitting process there is no set up required. TableCurve saves you precious time because it takes the endless trial and error out of curve fitting.

**Fit User Defined Equations**

Up to 15 user-defined equations can be entered and ranked along with the built-in equations. These specialized models can contain most mathematical constructs, including special functions, series convergence and conditional statements, differentiations, integrations, and parameter constraints. And, unlike most curve fitting programs, TableCurve 2D’s user-defined functions are compiled so custom curve fitting can be performed quickly, at nearly the speed as with the built-in equations. You can also add up to 100 external C or FORTRAN language functions to the TableCurve 2D equation set. These equations and constraints can be of unlimited complexity.

**Accurately Extrapolate Any Data Set**

Increase the accuracy of your predictions with state-of-the-art AR (Autoregressive) procedures that offer the means to effectively extrapolate any data set. Select from any one of the 9 different procedures for extrapolating your data – 3 to predict ahead, 3 to predict earlier data, and 3 that predict in both directions. Of these algorithms, six offer in-situ noise removal using advanced SVD and Eigendecomposition methods.

**All the Tools you Need to Visually Discover your Best Model**

**Graphically Review Curve Fit Results**

Once your XY data have been fit, TableCurve automatically sorts and plots the fitted equations by the statistical criteria you select (r2, DOF adjusted r2, Fit Standard Error or the F Statistic). Graphically review the fitted results as you scroll through the equation list. A residuals graph as well as parameter output are generated for each fitted equation. Add confidence and prediction intervals to the graph to detect outliers in your data. Data, statistical and numeric summaries are also available from within the Review Curve Fit window so you can further analyze fit results. TableCurve gives you all the information you need to discover the model that best meets your requirements for the ideal fit.

Compare Models Using Meaningful Numeric Information

Data, statistical and precision summaries are available so you can further analyze fit results. These summaries can be simultaneously displayed and are automatically updated when a different equation is selected for review. Evaluation option with automated table generation, includes function, derivatives, roots and cumulative area.

Effectively Manage Complex Data Sets

TableCurve 2D offers state-of-the-art smoothing and denoising techniques to remove the noise in your data. Select from a total of 6 smoothing/denoising algorithms. Of special importance is the Eigendecomposition Denoising, a non-parametric procedure where separation is based on signal strength. For Fourier Denoising, TableCurve 2D offers a data taper to minimize spectral leakage and the means to filter either by the magnitude of FFT channels or by frequency threshold. Inspect analytic derivatives for all built-in equations, as well as all of the smoothing procedures. Mask outliers and refit your data. With TableCurve 2D, its all so easy!

Precisely Model Exotic Data Sets

For those rare equations that cannot be adequately managed by a parametric model, TableCurve 2D offers three non-parametric estimation procedures. The Spline, Smoothing Spline and Local Regression options offer true state-of-the-art algorithms. For example, there are seven different spline algorithms, including two least-squares minimizations and the non-uniform n.

Flexible Output Options

With TableCurve you can preview your graph and output publication-quality graphs in several different configurations. You can also produce files containing data and equations in Lotus, Excel, ASCII, Harvard Graphics and SigmaPlot formats. TableCurve 2D can speed up your programming by generating actual function code and test routines for all fitted equations in FORTRAN, C, Basic, Pascal and VBA for Excel.

Maximize Your Productivity With Automation

Save time with the new unattended batch processing capability to automatically process a large number of data sets – no programming required! With this integrated automation capability, you can analyze a large number of data sets while you are away from your PC! TableCurve 2D automatically generates the output for each data set. The output can be written to an MS Word (or generic RTF) file for all graphs and numeric summaries, and to MS Excel for numeric data. The automation capability is available for all of TableCurve 2D’s major procedures.

## Product Features

The sections below explain the many useful features within TableCurve 2D:

**Interface**

- Full 32-bit performance
- Multitasking with 17 background thread curve-fit options
- Drag and drop files for immediate fitting
- Fully customizable 2D graphs
- Smooth bitmap rendering of graphs
- XY meter in graphs displays cursor coordinates
- Global Reset options to undo all changes made to the data table

**Data Input**

- Up to 65,536 points in data table
- 16.4 million points can be filtered into table using averaging digital import filter
- File types: ASCII, Excel (all versions through Excel 2000)*, Lotus, Quattro Pro, SigmaPlot, SPSS, dBase, DIF, Binary
- Graphical XY column selection
- Weights or standard deviations optionally assigned
- Compare up to five data sets
- Automatically process replicate data sets

**Data Management**

- Real-time smoothing with FFT, Loess, Savitzky-Golay, Gaussian Convolution, Eigendecomposition and Kaiser-Bessel time-domain smoothing* procedures
- Real-time Fourier domain editing with exact-N FFT; data tapering
- Eigendecomposition filtering to isolate components based on signal strength
- Graphical and numerical sectioning; graphically enable or disable data points
- First and second order analytic derivatives for all built-in equations*
- Multiple curve-fit references on a single graph (up to 4 references)*
- Apply calculations to X, Y and Weight values
- Spreadsheet-like data editing with optional graphing of data as they are entered

**Output and Export**

- Publication quality printed graphs with full print preview
- Image formats include bitmaps, metafiles, enhanced metafiles and device independent bitmaps
- File formats include ASCII, Excel, Lotus, SigmaPlot and Harvard Graphics Professional reports with MS Word or RTF export that includes graphs and numeric results; available for all major procedures
- Graphs can be half or full page, landscape or portrait mode all in a single document

**Equation Discovery and Curve Fitting**

- 3,656 built-in equations
- 3,205 mixed basis function linear
- Even order and half order polynomials
- 18 Chebyshev polynomials
- 17 High-precision polynomials (fitted to 38 digits precision)
- 10 Fourier-series polynomials
- 57 standard, ln x, sqrt even, y-transformed, even order, and half order rationals
- 17 Chebyshev rationals
- 17 High-precision rationals (fitted to 38 digits precision)
- 5 Fourier-series rationals
- 36 Constrained (no singularity) non-linearly fit SVD rationals
- 74 nonlinear peak equations
- 29 nonlinear transition equations
- 58 nonlinear kinetic equations
- 13 nonlinear waveform and miscellaneous equations
- Rapid searching, sorting and filtering of equations
- User customizable equation sets
- Full control of fit process, including goodness of fit criteria, matrix methods, minimization, and other options
- AI Expert for automatic selection of peak, transition or kinetic models to be fitted
- Three robust fitting methods available for all non-linear equations, user functions and external C/Fortran functions

Prediction Methods

- State-of-the-art Autoregressive procedures for forward/backward predictions/extrapolations.
- 9 different procedures – 3 to predict ahead, 3 to predict earlier data, and 3 that predict in both directions.
- In-situ noise removal using advanced SVD and Eigendecomposition methods
- View complex roots, order selection criteria, singular values, residuals and numeric summaries
- AR filter code generation is supported in all languages, plus export filter to disk

**User-Defined Functions (UDFs)**

- UDF editor with push button help for inserting functions
- UDFs automatically compiled for speed
- Up to 15 UDFs can be fit at one time, each with up to 10 adjustable parameters
- Graphical UDF adjustment procedure for refining starting estimates
- UDFs can be saved as libraries

**External 32-bit DLL User Defined Functions**

- Up to 100 external Fortran or C fitting functions with optional constraints
- Fortran requires MS Fortran Powerstation v4+ C requires MS Visual C++ v4+ or Borland C++ v4.5+
- Select all, none or specific equations

**Non-Parametric Fitting**

- Spline estimation, including least-squares B-splines and NURBS (non-uniform rational B-splines)
- Smoothing spline estimation, including first through sixth derivatives
- Local regression estimation

**Estimation and Interpolation**

- Eight spline procedures, 3 interpolation and 5 smoothing including NURBS and least-squares B-splines with fixed, optimized, or user-specified knots
- Local regression spline estimation; 1st and 2nd derivatives
- Savitzky-Golay spline estimation; accurate derivatives orders 1-8; filter coefficients
- Fourier Interpolation; derivatives orders 1-4;
- Directly specify and plot output for all Estimation procedures

**Curve-Fit Analysis and Output**

**Numerical**

- Evaluation option with automated table generation, includes function, derivatives, roots and cumulative area
- Full numeric and statistical summary, including coefficients, standard error, confidence limits, ANOVA, goodness of fit, measured function and derivative minima and maxima, and poles reported for rational functions
- Data summary with predicted values, residuals and confidence/prediction limits
- Precision summary and term significance analysis

**Graphical**

- Curve-fit graph with zoom-out, customizable layout, labels, grids, scaling, points, font, titles and resolution
- Confidence/prediction intervals (90, 95, 99, 99.9 and 99.99 percent)
- Error bars for replicate-based data
- Residuals graphs, separate or on Y2 axis of curve-fit graph, including bar graphs, histogram, and stabilized normal probability plot for assuring normality of residuals
- Copy numerical data for any graph to clipboard as a spreadsheet block
- Latent point processing enables disabling of outliers during equation inspection

## Automation

TableCurve 2D Version 5 can fit multiple data sets automatically in a batch-processing mode. The data source is either an Excel spreadsheet (or multiple spreadsheets) or an instrument via a custom DLL.

In addition to curve fitting, the Automation feature is available for other TableCurve features such as noise filtering, smoothing, local regression, splines and autoregressive modeling and prediction.

**Code Generation**

- Fortran 90, Fortran 77, C (64 and 80 bit), Visual Basic, QBasic and Pascal
- Function code or function code with full test routines
- Available for all built-in equations

#### Integrated Automation

- Batch processing for automatically processing large number of data sets unattended; available for all major procedures
- Multiple data sets in an Excel spreadsheet processed with the ease of single data set
- Stream reports directly to MS Word 95/97/2000 or RTF format
- DLL support for writing external data acquisition interface
- Professional DLL automation interface for instrument manufacturers

## Save Weeks of Tedious Data Analysis Chores

Many researchers spend hours and even weeks to discover the ideal model that describes their data. This is because they use tools that require trial and error curve fitting. TableCurve 2D is the first and only program that completely eliminates endless trial and error by automating the curve fitting process.

TableCurve 2D’s powerful curve fitter automatically fits 3,665 built-in equations from all disciplines to find the one that provides the ideal fit – instantly! What could once take days of tedious work now takes minutes, with much more powerful results.

**Solve Complex Science and Engineering Problems Faster**

- Optimized product and process performance
- Quickly calibrate sensors
- Understand complex chemical kinetics
- Reduce empirical data to a simple equation
- Stabilize systems with dynamic feedback
- Perform any general curve fitting application

**Powerful and Yet Easy To Use**

Once the fit is complete, TableCurve 2D presents you with a statistically ranked list of candidate equations. You are given all the information you need to choose the equation that best meets your requirements for the ideal fit. Once you have selected your best fit equation, output high – quality function code, or generate professional reports with publication – quality graphs. No other fitting program offers this much versatility and power, and with unbelievable ease. Users consistently comment that “out of the box, without reading the instructions” TableCurve 2D is highly intuitive, easy to use, and remarkably simple to learn.

## Applications

#### Equations For Every Application

**Optimize Process Control**

Quite often, substantial cost savings can be realized by optimizing process control parameters. Since TableCurve 2D automatically reports the minima and maxima of both fitted functions and derivatives, process optimization is often a simple matter of using a single number within the report.

**Generate Calibration Curves**

TableCurve 2D excels for applications in calibration science. Whether you need to calibrate a flow meter, wind tunnel measurements or satellite instrumentation signals, TableCurve 2D can furnish the ideal parametric model.

**Fit Tabulated Data**

Frequently, engineers and scientists need to convert tables of data found in handbooks or journals into a simple equation, often for use in software or microcode. TableCurve 2D is capable of producing equations that preserve all or most of the accuracy present within the tabulated data. TableCurve 2D even furnishes a Precision Summary for instrument designers who must use fixed point math in their microprocessors.

**Create Black Box Models**

At times, an engineer or scientist wants to study a highly complex process, such as human biochemistry, where underlying models are poorly defined. For such instances, TableCurve 2D can provide important insights into the subtle mechanisms in play. A successful model may suggest further experiments in pharmacokinetics, high energy physics or crystal chemistry.

**Produce Equations for Complex Modeling or Monte-Carlo Simulations**

TableCurve 2D can be used to convert empirical data from hundreds of sources into simplified equations. These equations can then be input into complex models or Monte-Carlo simulations to simulate the effects of hundreds or even thousands of variables upon a given model.

**Interpolate Data**

There is no simpler, faster or more automated way to interpolate data than with TableCurve 2D. Data tables or individual interpolated points are effortlessly generated for even the most demanding data sets.

## User Quotes

## Product Uses

#### Below are examples of TableCurve used in various fields and methods

**Analysis of Regional Pulmonary Ventilation**

An example is the analysis of regional pulmonary ventilation of a 133Xe airway bolus using a scintillation camera. Since the bolus of xenon initially washes into the lung and subsequently washes out, we would expect one of the TableCurve intermediate kinetic functions to fit the data. See the TableCurve PDF nonlinear equation documentation for description of the intermediate functions.

They represent measurement from the middle of three compartments. In our case the middle compartment is the lung with the airway and recirculation-via-organ-systems being the first and third compartments, respectively. Six of ten measured regions are shown in Figure 1. We want to fit all ten data sets as a batch and place the results in a Word file.

**Setting Up Automation**

This is a two step procedure:

- select options to fit the first data set
- use these options for Automation to fit all data sets

**Fit the First Data Set**

You fit the first data set exactly the same way you normally use TableCurve except in this case we are going to restrict the fitting to the Kinetic equations. Open TableCurve and Import the file regional 133Xe washout.xls that you have downloaded.

The Select Columns dialog, Figure 2, shows the data in columns A K of the Excel worksheet. This data is in X Many Y format with the X data in column A and the ten Y data sets in columns B – K.

Figure 2. Select Columns dialog with time data in column A and ten regional scintillation measurements in columns B through K.

Select column A and B to be the X and Y data. At the Process menu select Edit Custom Equation Set. For each tab select Clear to deselect the equations except for the Kinetic tab – select all Kinetic equations. The dialog for the Kinetic tab is shown in Figure 3.

Figure 3. Kinetic tab of the Custom Equations dialog. All functions are selected. All functions in all other tabs are deselected.

Click the Fit button shown in Figure 3 to fit these equations to the data in columns A and B of the Excel worksheet. Click Graph Start to view the Review Curve Fit window and equations. The equation list shows the highest ranked equation (by either R2 or F-Statistic) is “Intermed11 c < d(a,b,c,d)” which is what we expected. For this data set the Equil111 equations are probably not significantly different from the Intermed11 equations.

Figure 4. Equation list ranked by the F-statistic. The Intermed11 equations fit best.

**Use Automation to Fit All Data Sets**

We will now use Automation to fit all ten data sets with the equation selected to be the best during the fit of the first data set (Intermed11 c < d(a,b,c,d)) and place the results in a Word document. The Automation icon is located at the bottom of the icon collection of the Review Curve-Fit window. Click on it to obtain the Automation dialog.

Figure 5. The Automation dialog. The data source is the Excel file ‘regional 133Xe washout.xls’. The Word output file is ‘133Xe washout output.doc’. Two output options, Curve-Fit Graph and Numeric Summary, are selected to view the curve fit results and obtain numeric parameter values for the ten data sets.

Our data source is an Excel file and our data is located in the first worksheet Single X (XY in cols AB, AC, AD,…) format. These are the options selected in the Data Source group box in Figure 5.

Select the options in the Output group box in Figure 5 to put the ten data set curve fit results in the Word document ‘133Xe washout output.doc’.

Click OK to curve fit the ten data sets with the Intermed11 c < d(a,b,c,d) equation.

**Results of the Automated Fit**

Graphs for the fit to each data set and the numerical fit results are placed into the Word document. The results for the second data set are representative and are shown in Figure 6.

Figure 6. Automated results in the Word document for the second data set.

**Rank 1 Eqn. 8129 Intermed11 c < d ( a,b,c,d)**

r2 Coef Det | DF Adj r2 | Fit Std Err | F-value |

0.9860725867 | 0.9854922778 | 0.0283784161 | 2289.2272163 |

Parm | Value | Std Error | t-value | 95% | Confidence Limits | P>|t| |

a. | 0.012485740 | 0.009232193 | 1.352413322 | -0.00583761 | 0.030809087 | 0.17939 |

b. | 2.635130203 | 0.450727827 | 5.846389021 | 1.740560240 | 3.529700167 | 0.00000 |

c. | 0.058002827 | 0.007593421 | 7.638563579 | 0.042931989 | 0.073073665 | 0.00000 |

d. | 0.106282625 | 0.014492888 | 7.333433025 | 0.077518254 | 0.135046997 | 0.00000 |

An excellent fit is obtained with and F ratio > 2000. Of physiological interest, the clearance constant (c parameter) is 0.058 sec-1. For all data sets this clearance rate ranges from 0.028 to 0.065 sec-1 with corresponding time constants of 35 (apical lung region) to 15 seconds (basal lung region).

**Processing TableCurve Results**

The Word file is an excellent report format but if you want to continue to process the results of TableCurve Automation then use the Excel output option.

Figure 7. Options to place TableCurve Automation results into an Excel file.

The options shown in Figure 7 will place the fit results of the ten data sets into ten worksheets in the Excel file “133Xe washout output.xls”. You can then write a simple VBA macro in Excel to process these results. For example, you might create a table of lung clearance constant (c parameter) results for the ten data sets as a function of distance from lung apex to base.

[/toggle] [toggle border=’2′ title=’Measurement of Oil Droplet Size and Distribution’]

TableCurve 2D and SigmaScan Pro were used to characterize the size distribution of oil droplets. SigmaScan measured the oil droplet radius and TableCurve^{®} found the Weibull function to best characterize their size distribution. Oil droplets, suspended in a fluid column were imaged using conventional CCD camera technology and a PC compatible frame grabber board.

The image shown in **Figure 1** is: an image of oil droplets suspended in a fluid column.

**Figure 2:** Image after contrast enhancement. The image was calibrated using a two point calibration from the Calibrate, Distance and Area option in the Image menu. The contrast was enhanced using the Histogram Stretch procedure (Image, Intensity menu) so that operators could better visualize the oil droplets (the Old Start line end point was dragged with the mouse to intensity 192 which stretches the light gray (192) to white (255) range over then entire 0 – 255 range).

The enhanced image is shown in Figure 2.

** **

** **

**Figure 3:** Selecting the oil droplets using intensity thresholding. Intensity Thresholding the image in the intensity range 0 – 140 using the Image, Threshold option selected the darker oil droplets.

The selected oil droplets are shown in the red overlay plane in Figure 3.

**Figure 4:** Using the Fill Holes procedure to eliminate open areas in the oil droplets. Due to the surface reflections, intensity thresholding will not select all pixels in some of the oil drops. The Fill Holes feature in the Image, Overlay Filters dialog was used to allow accurate droplet area measurements to be made.

This fills the holes in the droplets as shown in Figure 4.

**Measurements and Results**

The objects on the red overlay plane were then counted and the parameters perimeter, area, ferret diameter, shape factor, compactness and number of pixels measured using the Measure Objects option in the Measurements menu. These measurements were selected from the list in the Measurements tab in the Measurements Settings dialog.

A macro was written to compute the circular radius of each droplet using the equation R = (A/pi)^0.5 and the results placed into the worksheet. A histogram of droplet radius from 0 to 10 microns was also computed. An ASCII file report was generated and formatted in Excel.

Perimeter | Area | Feret Diameter | Shape Factor | Compact | # Pixels | Radius | Size Range | Number of Droplets |

9.57 | 6.99 | 2.98 | 0.96 | 13.11 | 16 | 1.49 | 0 | 0 |

1.87 | 0.44 | 0.75 | 1.57 | 8 | 1 | 0.37 | 1 | 51 |

9.96 | 6.11 | 2.79 | 0.77 | 16.22 | 14 | 1.39 | 2 | 37 |

66.91 | 63.31 | 14.44 | 0.46 | 27.41 | 374 | 7.21 | 3 | 30 |

15.57 | 17.03 | 4.66 | 0.88 | 14.23 | 39 | 2.33 | 4 | 13 |

31.91 | 34.5 | 6.63 | 0.43 | 29.51 | 79 | 3.31 | 5 | 6 |

25.53 | 19.65 | 5 | 0.38 | 33.16 | 45 | 2.5 | 6 | 2 |

3.19 | 0.87 | 1.05 | 1.08 | 11.66 | 2 | 0.53 | 7 | 3 |

1.87 | 0.44 | 0.75 | 1.57 | 8 | 1 | 0.37 | 8 | 1 |

14.63 | 15.28 | 4.41 | 0.9 | 14.01 | 35 | 2.21 | 9 | 0 |

1.87 | 0.44 | 0.75 | 1.57 | 8 | 1 | 0.37 | 10 | 0 |

14.08 | 10.04 | 3.58 | 0.64 | 19.75 | 23 | 1.79 | ||

14.86 | 11.35 | 3.8 | 0.65 | 19.45 | 26 | 1.9 | ||

19.53 | 19.21 | 4.95 | 0.63 | 19.85 | 44 | 2.47 | ||

14.08 | 10.92 | 3.73 | 0.69 | 18.17 | 25 | 1.86 |

The histogram data in the last two columns was copied into TableCurve 2D. All peak functions were selected from the Custom Equation dialog and the Weibull distribution found to fit the data best. The TableCurve® graph of these results is shown in Figure 5 below:

[/toggle] [toggle border=’2′ title=’USA Young America Optimized With TableCurve 2D®’]

PACT95, the coalition that designed the fastest American yacht in the 1995 America’s Cup race, Young America, chose TableCurve 2D to help optimize their design because it is the only software that, in seconds, searches through thousands of equations to describe difficult-to-model empirical data.

Taking data samples from numerous sources, including wind tunnel tests, tank testing, and data from many other types of experiments, TableCurve® was able to convert vast amounts of empirical data into simplified curve fits. These algorithms were then input into a Velocity Prediction Program (VPP) that balances the equations of motion while looking for the fastest boat.

[/toggle] [toggle border=’2′ title=’Explicit Function Approximation’] A closed form expression for F cannot be obtained for the Baker and Lonsdale equation. However, since the expression for F does not contain the parameter k, it can be numerically ‘inverted’. The numerical inversion was done with TableCurve 2D®.

A closed form expression for F cannot be obtained for the Baker and Lonsdale equation (first equation in the Table). However, since the expression for F does not contain the parameter k, it can be numerically ‘inverted’. The numerical inversion was done with TableCurve 2D® using the following procedure. First the expression for F on the left side of the equation was evaluated for 1000 equidistant F values from 0 to 100.

The columns containing the expression for F (which equals kt) and F were then reversed so that F is now the Y variable and kt is the X variable. All equations in TableCurve were then fit to this X,Y data set and the equations ranked using the F statistic. The best fitting equation found (Fstat = 2.7×1014) was the following rational polynomial in fractional powers of x.

where the coefficients are

a = 2.5788672e-6

b = -3.4434044

c = 244.94883

d = 3.9105658

e = -976.78997

f1= -1.5002823

g = 1407.9333

h = 0.039306878

i = -862.63205

j = 0.0091845726

k1=187.88278

In this equation F is an explicit function of x (=kt) so it is used in the nonlinear curve fitter to estimate the parameter k. An analysis of the residuals found the maximum absolute difference between the actual F and approximate F to be 0.0003 (at F= 99.9). The maximum absolute percentage difference was only 0.002% (at F = 0.1).

## System Requirments

#### TableCurve 2D System Requirements

#### Hardware

- 486 Processor or higher
- 16 MB RAM required (32 MB or more recommended)
- 10MB hard disk space

#### Software

Windows 10, 8.x, 7, Windows Vista, 95, 98, NT and XP (32 bit)