Curve Fit Models


There are several different models available for curve fitting.

Straight Line

The straight line fit is calculated by choosing the line that minimizes the least square sum of the vertical distance d, of all the selected markers (see picture below) by using the following equation:

curve_linear_regression_eq.png

where a is the intercept and b is the slope.

For example, you could plot days along the X-axis and have one marker for each day. The distance between the markers along the X-axis is the same, thus making straight line fit appropriate.

 

images/dsb_straight_line_fit2.png

Logarithmic

The logarithmic fit calculates the least squares fit through points by using the following equation:

curve_logarithmic_eq.png

where a and b are constants and ln is the natural logarithm function. This model requires that x>0 for all data points.  Spotfire uses a nonlinear regression method for this calculation. This will result in better accuracy of the calculation compared to using linear regression on transformed values only.

Exponential

The exponential fit calculates the least squares fit through points by using the following equation:

curve_exponential_eq.png

where a and b are constants, and e is the base of the natural logarithm.

Exponential models are commonly used in biological applications, e.g., for exponential growth of bacteria. Spotfire uses a nonlinear regression method for this calculation. This will result in better accuracy of the calculation compared to using linear regression on transformed values only.

Power

The Power fit calculates the least squares fit through points by using the following equation:

curve_power_eq.png

where a and b are constants. This model requires that x>0 for all data points, and either that all y>0 or all y<0. Spotfire uses a nonlinear regression method for this calculation. This will result in better accuracy of the calculation compared to using linear regression on transformed values only.

Logistic Regression

The logistic regression fit is a dose response ("IC50") model, also known as sigmoidal dose response. The four parameter logistic model is the most important one.

Dose-response curves describe the relationship between response to drug treatment and drug dose or concentration. These types of curves are often semi-logarithmic, with log (drug concentration) on the X-axis. On the Y-axis one can show measurements of enzyme activity, accumulation of an intracellular second messenger or measurements of heart rate or muscle contraction.

Log10-transformed X-values:

The logistic regression on logged X-values fit uses the following equation:

curve_logistic_regression_on_logged_x_eq.png

The LoggedX50 value is interpreted as the Log10(X50). For example, if the H30+ concentration at IC50 has a pH of 3, then the LoggedX50 = -3.

Note: With this model, it is the logged X50 values that are estimated and not the actual X50.

 

Non-logarithmic X-values:

The logistic regression fit when not assuming logged X-values uses the following equation:

curve_logistic_regression_eq.png

where min and max are the lower and upper asymptotes of the curve, Hill is the slope of the curve at its midpoint and X50 is the x-coordinate of the inflection point (x, y). This model requires that x>0 for all data points and that you use at least four records to calculate the curve.

curve_logistic_regression_curve.png

Polynomial

The polynomial curve fit calculates the least squares fit through points by using the following equation:

curve_polynomial_eq.png

where a0, a1, a2, etc., are constants.

See also:

Lines and Curves