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1. Simple statistical measures • Estimating Mean and Variance • Histogram • Simple Linear Regression 2. Cartographic modeling Simple statistical measures • Estimating Mean and Variance • Histogram • Simple Linear Regression Estimating Mean and Variance The formulae given before to calculate the mean and variance requires to know all the probability mass function or probability density function for all possible measurements (or the entire population). In reality we often do not know them a priori because the size of the population may be very big or infinite. The way to get information about the mean and variance for a population is to take a sample and infer from there. Given a sample of x1,x2, …, xn randomly taken from a population, the population mean and variance are estimated from the sample as in the following 1 n ˆ xi n i 1 n 1 2 ˆ ˆ 2 ( x ) i n 1 i 1 Why Squared? The variance is the average squared difference of an individual value from the population mean. Why squared? Example cases: Suppose we want to evaluate two GPS receivers for their accuracy. We take the receivers out to measure the distance between two points that are exactly 100.00 meters. We take 5 reading using each GPS and the following are the distance: Receiver1: 103, 97, 105 93 102 Receiver 2: 101, 100, 99, 98 103 They both give an average distance of 100.00 m. Are there difference in their accuracy? What are the variances for each receiver? Answer Receiver 1: 1 N 1 2 s ( x x ) (103 100) 2 (97 100) 2 (105 100) 2 (93 100) 2 (102 100) 2 i N 1 i 1 4 2 1 (3) 2 (3) 2 (5) 2 (7) 2 (2) 2 24.0 4 Receiver 2: 1 N 1 2 s ( x ) (101 100) 2 (100 100) 2 (99 100) 2 (98 100) 2 (103 100) 2 i N 1 i 1 4 2 1 2 (1) (0) 2 (1) 2 (2) 2 (3) 2 3.75 4 Histogram The mean and variance tell us the overall magnitude and variation of data in a sample. An graphic view about the same information is called a histogram. A Histogram is a graph that shows the frequency for all the data observed in a sample. For Example, the following table shows the frequency distribution for a Geog370 midterm. The graph to the right is the histogram showing the frequency distribution. Frequency Ratings 80s 70s 60s 50s 20s Frequency 10 14 7 3 1 14 10 7 3 20 50 60 70 80 Grades Histogram Shapes Bell Shape Bimodal Mode: value with highest frequency Range: largest value-smallest value Skewed Random The definition of relative frequency probability tells that if we increase the number of observations for the random variable, the relative frequency of the histogram gets increasingly closer to its probability distribution. Histogram Landsat TM 1993 Landsat ETM 2002 Mode Mode is defined as the peak of the frequency distribution or the most frequent class. Median The median of a set of n observations, x1, x2, …, xn is defined to be the central value when the observations are arranged in order of magnitude. If there is an even number of observations, the median value is the midpoint between the two center observations Standard Deviation Square root of variance, which express the variability in measurement of the original units. Skewness: measure the degree of asymmetry about the mean of the data. Simple Linear Regression In the previous slides, we only focus on one random variable. In many applications, we often work with a pair of variables. For example the distance travels and the time spent driving; one’s age and height. Generally, there are two types of relationships between a pair of variable: deterministic relationship and probabilistic relationship. Deterministic relationship s s0 vt S: distance travel S0: initial distance v: speed t: traveled distance slope S0 v intercept time Probabilistic Relationship In many occasions we are facing a different situation. One variable is related to another variable as in the following. height age Here we can not definitely to predict one’s height from his age as we did in s s0 vt Linear Regression Statistically, the way to characterize the relationship between two variables as we shown before is to use a linear model as in the following: y a bx Here, x is called independent variable y is called dependent variable is the error term y a is intercept b is slope Error: b a x Least Square Lines Given some pairs of data for independent and dependent variables, we may draw many lines through the scattered points y x The least square line is a line passing through the points that minimize the vertical distance between the points and the line. In other words, the least square line minimizes the error term . Least Square Method For notational convenience, the line that fits through the points is often written as yˆ a bx The linear model we wrote before is y a bx If we use the value on the line, ŷ , to estimate y, the difference is (y- ŷ) For points above the line, the difference is positive, while the difference is negative for points below the line. y yˆ a bx ŷ (y- ŷ) Sum of Squares error For some points, the values of (y- ŷ) are positive (points above the line) and for some other points, the values of (y- ŷ) are negative (points below the line). If we add all these up, the positive and negative values can get cancelled. Therefore, we take a square for all these difference and sum them up. Such a sum is called the Error Sum of Squares (SSE) n SSE ( y yˆ ) 2 i 1 The constant a and b is estimated so that the error sum of squares is minimized, therefore the name least squares. Estimating Regression Coefficients If we solve the regression coefficients a and b from by minimizing SSE, the following are the solutions. n b ( x x )( y y ) i i 1 i n 2 ( x x ) i i 1 a y bx Where xi is the ith independent variable value yi is dependdent variable value corresponding to xi x_bar and y_bar are the mean value of x and y. Interpretation of a and b The constant b is the slope, which gives the change in y (dependent variable) due to a change of one unit in x (independent variable). If b> 0, x and y are positively correlated, meaning y increases as x increases, vice versus. If b<0, x and y are negatively correlated. y y a a b<0 b>0 x x Correlation Coefficient Although now we have a regression line to describe the relationship between the dependent variable and the independent variable, it is not enough to characterize the relationship between x and y. We may see the situation in the following graphs. y (a) y x (b) x Obviously the relationship between x and y in (a) is stronger than that in (b) even though the line in (b) is the best fit line. The statistic that characterizes the strength of the relationship is correlation coefficient or R2 R Square Regression Sum of Squares y n SSR ( yˆ i y ) 2 ŷ i 1 Total Sum of Squares n SST ( yi y ) 2 i 1 R2 SSR SST R square indicates the percent variance in y explained by the regression. y An Simple Linear Regression Example The followings are some survey data showing how much a family spend on food in relation to household income (x=income in thousand $, y=$ on food) x y 6.5 81 4 96 2.5 93 7.2 68 8.1 63 3.4 84 5.5 71 sum 37.2 556 mean 5.31429 79.4286 slope -5.2071 intercept 107.101 SST 953.714 SSR 706.834 SSE 246.881 SST+SSR 953.715 R-square 0.74114 x-x_bar 1.185714 -1.31429 -2.81429 1.885714 2.785714 -1.91429 0.185714 y-y_bar (x-x_bar)(y-y_bar) 1.571429 1.863265306 16.57143 -21.77959184 13.57143 -38.19387755 -11.4286 -21.55102041 -16.4286 -45.76530612 4.571429 -8.751020408 -8.42857 -1.565306122 -135.7428571 (x-x_bar)^2 1.40591837 1.72734694 7.92020408 3.55591837 7.76020408 3.6644898 0.0344898 26.0685714 y_hat 73.254325 86.2722 94.082925 69.60932 64.922885 89.39649 78.461475 (y-y_bar)^2 (y_hat-y_bar)^2 (y-y_hat)^2 2.46938776 38.12130132 59.99548121 274.612245 46.83527158 94.63009284 184.183673 214.7501205 1.172726556 130.612245 96.41767056 2.589910862 269.897959 210.4148973 3.697486723 20.8979592 99.35942913 29.12210432 71.0408163 0.935272739 55.67360918 953.714286 706.8339631 246.8814117 NDVI and Precipitation Relationships A: 12 Apr-2 May 1982 B: 5 to 25 Jul 1982 C: 22 Sep to 17 Oct 1982 D: 10 Dec 1982-9Jan 1983 Expansion and contraction of the Sahara • How the various properties of a location are related is an important aspect of the nature of geographic data Y=f(x1,x2,x3,…,xk) Y: the value of individual properties in a city X1:floor area X2: distance to parks X3: distance to schools …. Spatial autocorrelation • “Spatial autocorrelation is determined both by similarities in position, and by similarities in attributes” • The Tobler Law: everything is related to everything else, but near things are more related than distant things. Spatial autocorrelation • Positive spatial autocorrelation: Features that are similar in location are also similar in attributes • Negative spatial autocorrelation: Features that are similar in location are dissimilar in attributes • Zero autocorrelation Features are independent of location • Cartographic modeling “A cartographic model provides information through a combination of spatial data sets, functions, and operations” Functions and operations: reclassification, overlay, interpolation, terrain analyses, buffering and other proximity functions. Cartographic models: an example • Suitability analyses are perhaps the most common examples of cartographic models. • Suitability analyses rank land according to their utility for various uses. Cartographic models: an example • Suitable sites: (a) near lakes (b) near roads (c) not wetland Cartographic models: an example • Data Lakes, roads, and hydric status • Spatial operations Buffering, reclassification, and overlay Flowcharts: Cartographic models: weightings among criteria Criteria for a home-site selection: 1) Slopes should not be too steep 2) Sites should be far enough from a main road to offer some privacy, but not so far as to be isolated. 3) ? The conversion from a qualitative to quantitative specification? Weightings among criteria • How to combine distinct criteria? - Overlay - Addition We must choose how to weight one layer relative to another. Home-site selection: - How important is isolation relative to other factors It is often difficult to assign the relative weights in an objective fashion • One methods of assigning weights is based on their importance ranking. 1. Rank the importance 2. Calculate the relative weights according to: