CALCULATIONS

---

Basic Calculations Used in Fractal Analysis

---

---

Standard Error

How is the standard error (SE) 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, is used as an approximation for the Dʙ:

m = (nΣSC - ΣSΣC)/(nΣS² - (ΣS)²)

 where S = log of scale or size,
C = log of count,
and n = number of sizes.

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

See Graphing Regression lines

---

Correlation

How is the correlation coefficient, 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 = Euler's e^y-int

y = AX^DB

Where for y = AX^DB, -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.

Generally, the CV measures variation in a set of data as the ratio of the standard deviation over the mean, but it is often multiplied by 100 or squared, depending on the usage.

In FracLac, it is used to calculate lacunarity, where it is abbreviated, λ. It is a measure of variation in pixel distribution for regular box counting and sliding box lacunarity.

CV² = [σ / μ]²=λ

---

HOME