Calculus Functions
This section contains general purpose calculus functions.
Area
Perform cumulative area-under-the-curve computation with two columns of data of equal length. The result will be returned in a column of the same length as the input columns.
Parameter Set
| Parameters | Description |
| Column X | X column for the area operation. Column numbers begin with zero |
| Column Y | Y column for the area operation. Column numbers begin with zero |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Area Under The Curve (AUC) calculations are used widely in the financial, medical, and engineering sciences. Some specific examples are measuring bloodstream absorption of drugs, strength of materials like steel, plastic, the total force of wind on a particular building shape, etc.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Column X = 1 Column Y = 0 Area Option = AGGRAGATE Result Column X Name = Result Column Y Name = Include Original Data = Yes |
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Chart Sample
This chart shows a strength of an alloy. The bottom plot area shows the effect of stress. The top plot area is an “area under the curve” calculation of the line in the bottom plot. It shows the strength of the alloy although it is less pliable.
Differentiation
Rate of change calculation. Perform cumulative differentiation computation with two columns of data of equal length. The result will be returned in a column of the same length as the input columns.
Parameter Set
| Parameters | Description |
| Column X | X column for the differentiation operation. Column numbers begin with zero |
| Column Y | Y column for the differentiation operation. Column numbers begin with zero |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Used to determine the rate of change – velocity, marginal cost, marginal revenue, etc, for a given process at a given point.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||||||||
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Column X = 0 Column Y = 1 Result Column X Name = Tons Result Column Y Name = Marginal Cost Include Original Data = Yes |
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Chart Sample
This chart shows the differentiation function (marginal cost) over the cost to produce a ton of cement. As the volume of cement production increases, the cost per ton goes down.
Curve Length
Perform cumulative computation of the length of a 2 dimensional curve assuming that the x and y points respectively are provided in two columns of equal length. The result will be returned in a column of the same length as the input columns.
Parameter Set
| Parameters | Description |
| Column X | X column for the curve length operation. Column numbers begin with zero |
| Column Y | Y column for the curve length operation. Column numbers begin with zero |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Measuring geographic borders, civil engineering, computer-aided-design, volatility.
Data Sample
| Input | Parameter | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Column X = 0 Column Y = 1 Result Column X Name = P1X Result Column Y Name = P1Y Include Original Data = Yes |
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Chart Sample
The chart below shows a comparison of two routes. The distance is cumulatively calculated, showing that “Proposed Route 1” is the shortest route (but not by much).
Surface Metrics
Compute data point count, curve length and surface area of a 2 dimensional curve assuming that the x and y points respectively are provided in two columns of equal length.
Parameter Set
| Parameters | Description |
| Column for X | X column for the surface metrics operation. Column numbers begin with zero |
| Column for Y | Y column for the surface metrics operation. Column numbers begin with zero |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
This is a summary function that provides curve length, point count, and surface area.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||
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Column for X = 0 Column for Y = 1 Include Original Data = Yes |
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Regression
Regression is performed in order to find a mathematical equation that closely represents a set of paired (x,y) data. This allows one to estimate y values for x points other than the ones present in the original data set. This module fits two columns of related data (x,y pairs) to an equation of the form y = f(x), where the function f(x) is determined by the type and order of regression selected. The result x column returns a new set of x values (the original set or a smoothened one). The result y column returns the function f(x) evaluated at the new x points. The new (x,y) pairs need not include the original ones.
Parameter Set
| Parameters | Description |
| Column X | X column for the regression operation. Column numbers begin with zero |
| Column Y | Y column for the regression operation. Column numbers begin with zero |
| Operation | Type of regression equation to which the data will be fit. POLYNOMIAL , EXPONENTIAL , HYPERBOLA , LOGARITHMIC, INVERSE, INVERSE_POWER, POLYEXP, POWER (Refer to Appendix for regression equation) |
| Degree | Polynomial degree applicable only to those regression equations that are expressed as a polynomial series (POLYNOMIAL, INVERSE, LOGARITHMIC, POLYEXP) |
| Smoothen | Total number of (x,y) pairs to be returned in the result columns. This number should be larger than the original data set. |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Useful in statistical surveys, polling, or other situations where a finite set of samples is used to determine the characteristics of the overall data set.
Data Sample
| Input | Parameter | Result | |||||||||||||||||||||
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Column X = 0 Column Y = 2 Operation = POLYNOMIAL Degree = 1, 3 or 5 Smoothen = 20 Result Column X Name = X1D1, X1D3, or X1D5 Result Column Y Name = Y1D1, Y1D3, or Y1D5 Include Original Data = Yes |
See Plot displayed below. This uses 3 Regressions on the data for 1,3 and 5 degrees |
Chart Sample
Fourier
Fourier regression is performed in order to find a mathematical equation that closely represents a set of paired (x,y) data that is periodic and can be well represented by a Fourier series. This allows one to estimate y values for x points other than the ones present in the original data set. This module fits two columns of related data (x,y pairs) to an equation of the form ( f(x) = b0 + b1cos(x) + b2sin(x) + b3cos(2*x) + b4sin(2*x) + …). The result x column returns a new set of x values (the original set or a smoothened one). The result y column returns the function f(x) evaluated at the new x points. The new (x,y) pairs need not include the original ones.
Parameter Set
| Parameters | Description |
| Column X | X column for the Fourier regression operation. Column numbers begin with zero |
| Column Y | Y column for the Fourier regression operation. Column numbers begin with zero |
| Order | For a selected order of n, the Fourier equation fit to the data will extend up to the terms including cos(nx) and sin(nx) |
| Smoothen | Total number of (x,y) pairs to be returned in the result columns. This number should be larger than the original data set. |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Useful in statistical surveys, polling, or other situations where a finite set of samples is used to determine the characteristics of the overall data set.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||
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Column X = 0 Column Y = 1 Order = 5 Smoothen = 30 Result Column X Name = For X Result Column Y Name = For Y Include Original Data = Yes |
See Plot displayed below |
Chart Sample
In the plot shown below, the original data is displayed along with the results of Fourier regression.
Filter
The filter module fits a set of periodic data to a Fourier series of selected order and filters out unwanted orders (e.g. remove noise from a signal). Two columns of related data (x,y pairs) are fit to an equation of the form ( f(x) = b0 + b1cos(x) + b2sin(x) + b3cos(2*x) + b4sin(2*x) + …). The result x column returns a new set of x values (the original set or a smoothened one). The result y column returns the function f(x) evaluated at the new x points with only terms of the desired orders included.
Parameter Set
| Parameters | Description |
| Column X | X column for the filter operation. Column numbers begin with zero |
| Column Y | Y column for the filter operation. Column numbers begin with zero |
| Filter | Filter type options include Low Pass (remove orders above second cut-off order), High Pass (remove orders below first cut-off order), Band Pass (only keep orders between first cut-off order and second cut-off order), Band Stop (remove orders between first cut-off order and second cut-off order), Dominant (keep number of dominant orders specified and remove the rest) |
| First Cut-off Order | Cut-off applicable to High Pass, Band Pass and Band Stop Filter options |
| Second Cut-off Order | Cut-off applicable to Low Pass , Band Pass and Band Stop Filter options |
| Dominant Orders | Number of dominant orders to be retained beginning from the most dominant to the least. The dominant orders are the ones for which the Fourier coefficients have the greatest magnitude |
| Order | For a selected order of n, the Fourier equation fit to the data will extend up to the terms including cos(nx) and sin(nx) |
| Smoothen | Total number of (x,y) pairs to be returned in the result columns. This number should be larger than the original data set. |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Digital signal processing, other fields where refinement of the data input is required.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||
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Column X = 0 Column Y = 1 Filter = Low Pass First Cut-off Order = 1 Second Cut-off Order = 1 Dominant Orders= Order = 5 Smoothen = 50 Result Column X Name = LPX Result Column Y Name = LPY |
See Plot displayed below |
Chart Sample
In the plot shown below, the original data is displayed along with the results of low-pass (order 1) and high pass (order 2) filtering.
Track Order
The Track Order module fits a set of periodic data to a Fourier series of selected order and calculates the magnitude of each order. Two columns of related data (x,y pairs) are fit to an equation of the form ( f(x) = b0 + b1cos(x) + b2sin(x) + b3cos(2*x) + b4sin(2*x) + …). The coefficients of the fit are used to calculate the magnitude of each order.
Parameter Set
| Parameters | Description |
| Column X | X column for the Track Order operation. Column numbers begin with zero |
| Column Y | Y column for the Track Order operation. Column numbers begin with zero |
| Order | For a selected order of n, the Fourier equation fit to the data will extend up to the terms including cos(nx) and sin(nx) |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
Helps determine the order that is the best fit for the input data. Useful in selecting the precise filter to be applied.
Data Sample
| Input | Parameter | Result | ||||||||||||||||||||||||||||||||||||
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Column X = 0 Column Y = 1 Order = 5 Result Column X Name = TOX Result Column Y Name = TOY Include Original Data = Yes
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See Plot displayed below |
Chart Sample
The following plot shows Track Order analysis performed on the above input data. The dominant orders in this case are 2 and 1.
Interpolation
Interpolation on a set of scattered (x,y) data is performed in order to estimate y values at intermediate x points. A piecewise fit on subsets of data points is performed with a spline or piecewise polynomial of selected degree. The result (x,y) pairs include the original data plus (x,y) pairs evaluated at the number of intermediate points specified by the smoothen parameter.
Parameter Set
| Parameters | Description |
| Column X | X column for the interpolation operation. Column numbers begin with zero |
| Column Y | Y column for the interpolation operation. Column numbers begin with zero |
| Operation | Spline type LINEAR, QUADRATIC or CUBIC indicating spline of degree 1, 2 or 3 respectively |
| Segment | Number of points per segment including both end points. Each consecutive pair of (x,y) points in the input data is treated as a segment. |
| Result Column X Name | Name of result column X |
| Result Column Y Name | Name of result column Y |
| Include Original Data | If “Yes” include the original dataset in the result table |
Applications
General line smoothing to show trends or a smooth distribution.
Data Sample
| Input | Parameter | Result | ||||||||||||
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Column X = 0 Column Y = 1 Operation = CUBIC Segment = 10 Result Column X Name = CX Result Column Y Name = CY Include Original Data = Yes |
See Plot displayed below |
Chart Sample
