This page outlines the basic calculations essential to calculating the DB and lacunarity with FracLac.

Calculations

Standard Error

How is the standard error calculated? SE=√ [(ΣC2 - bΣC - mΣSC)/(n-2)] where S = log of scale or size, C = log of count, n = number of sizes, b = y-intercept of the regression line, m = slope of the regression line  

Regression Line

How is the slope of the regression line calculated? The slope of the regression line, m, used for calculating the DB = m = (nΣSC - ΣSΣC)/(nΣS2 - (ΣS)2)  where S = log of scale or size, C = log of count, n = number of sizes, m = slope of the regression line

For other regression lines, S=the value along the x-axis, and y = the value along the y-axis.  

Correlation

How is the correlation (r2) for the regression line calculated? r2= [(nΣSC-ΣSΣC)/√ [(nΣS2-(ΣS)2)(nΣC2-(ΣC)2)] ]2 where S = log of scale or size, C = log of count, n = number of sizes, b = y-intercept of the regression line

Y-intercept

How is the y-intercept of the regression line calculated? y int = (ΣC-mΣS)/n where S = log of scale or size, C = log of count, n = number of sizes, m = slope of the regression line

Prefactor

How is the prefactor for the scaling rule calculated? The prefactor A: A = Eulers ey-int y = AXDB Where for y = AXDB, -DB = slope of the regression line and y-int = the y-intercept of the regression line 

CV

What is a CV?

CV stands for coefficient of variation = standard deviation/mean. CV2 = [σ/μ]2 In FracLac, it is used to calculate lacunarity It is a measure of variation in pixel distribution for regular box counting and sliding box lacunarity. It measures variation in a set of data and is calculated as the standard deviation over the mean for the data. It can be multiplied by 100 or squared, depending on the usage.