Basic Statistics

2. Basic Statistics#

Statistics on attributes
- Univariate statistics
- Multivariate statistics
- Inferential statistics

Measures of central tendency
- Mean, median, mode
Measures of dispersion
- Range, minimum, maximum
- Variance and standard deviation
  - The extent to which values vary from the mean
- Coefficient of variation (CV)

How two attributes are related to each other
Correlation Coefficient
- A (normalized) measurement of the covariance of two variables
- Has a value between -1 and 1
- Formula: \(r=\frac{\sum_{i=1}^{n}(y_{i}-\overline{y})(z_{i}-\overline{z})}{\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}\sqrt{\sum_{i=1}^{n}(z_{i}-\overline{z})^{2}}}\)

Variable 1 (y)	Variable 2 (zᵢ)	(yᵢ-ȳ)	(zᵢ-ž)	(yᵢ-ȳ)×(zᵢ-ž)	(yᵢ-ȳ)²	(zⱼ-ž)²
12	6	-20.11	-27.00	543.00	404.46	729.00
34	52	1.89	19.00	35.89	3.57	361.00
32	41	-0.11	8.00	-0.89	0.01	64.00
12	25	-20.11	-8.00	160.89	404.46	64.00
11	22	-21.11	-11.00	232.22	445.68	121.00
14	9	-18.11	-24.00	434.67	328.01	576.00
56	43	23.89	10.00	238.89	570.68	100.00
75	67	42.89	34.00	1458.22	1839.46	1156.00
43	32	10.89	-1.00	-10.89	118.57	1.00
SUM				3092	4114.89	3172.00

Calculation:
- Numerator: 3092
- Denominator: \(\sqrt{4114.89} \times \sqrt{3172.00} = 64.1474 \times 56.3205 = 3612.8136\)
- \(r = 3092 / 3612.8136 = 0.8558\)

Relationship between dependent and independent variable(s)
Ordinary least square regression
- Least square approach
- Minimize the sum of squared errors (residuals)
- Coefficients are calculated by solving a set of equations
- Equation: \(z = a * y + b\)
  - a — slope
  - b — intercept
Example: \(z = 8.8711 + 0.7514y\)
- \(r^2 = 0.7325\)
Goodness of fit
- Coefficient of determination
- Percent of variation in z explained by y
- \(r^2 = (\text{correlation coefficient})^2\)

Conventional correlation and linear regression
- Two variables measured on the same subjects
- Variables related by subjects
Geospatial correlation and linear regression
- Variables related by location (and time)
- Two variables at the same location
  - Example: temperature ~ elevation regression
- Geographically weighted regression
  - Two variables nearby a location
Spatial auto-correlation
- How a variable is related to itself in space (and time)
- Same attribute at two different locations

Where is the center of a geographic distribution
How features are distributed around their center
Why measure geographic distributions?
- Know the general location of geographic features
- Compare the distributions of different features/phenomena
- Track changes of distribution in time
Modified from one dimension (attribute) to two dimensions

Central feature
- The most centrally located feature
- Shortest total distance from all the other features
Median center
- Minimizing the sum of the distances to each point
- Less influenced by outliers
- No simple formula (iterative process)

Center influenced by the attribute at a location
Center of gravity
Formula: \(\overline{X}_w = \frac{\sum W_i X_i}{\sum W_i}, \overline{Y}_w = \frac{\sum W_i Y_i}{\sum W_i}\)
Example:
- Points: (5, 0), (2, 4), (5, 8) with weights 2, 4, 4
- Mean center (4, 4)
- Weighted mean center (4.4, 4)

How feature locations deviate from their mean center
Average distance to mean center
Two dimensional equivalent of standard deviation
Compactness (dispersed vs. clustered)
Formula: \(S_{xy} = \sqrt{\frac{\sum (X_i - \overline{X})^2 + \sum (Y_i - \overline{Y})^2}{N}}\)