Spatial Interpolation & Trend Surface Analysis Flashcards
Spatial interpolation background
- Data collection and analysis is expensive and infrequently collected
- Unknown values must be estimated from collected data that can be sampled
- Interpolation is procedure of estimating unknown values w/in area covered by existing observations
- x, y, and z are important
What can spatial interpolation be used for in GIS?
- Provide contours to display data graphically
- Calculate some property of the surface at a given point
- Calculate value for how good the model fits data (where it performs well vs. poorly)
Spatial interpolation is often used to aid what?
- Spatial decision making process both in physical and human geography
- ex. Mineral prospecting and hydrocarbon exploration
4 Types of Spatial Interpolation
- Global/Local
- Exact/Approximate
- Stochastic/Deterministic
- Gradual/Abrupt
Global Interpolators
- Determine a single function which is mapped across the whole region
- A change in one input value affects the entire map
Local Interpolators
- Algorithm is applied repeatedly to a small portion of total points
- A change in one input value only affects the result within the window
Exact Interpolators
- Surface passes through all points whose values are known
Approximate Interpolators
- Used when there is some uncertainty about the given surface values
Stochastic Method
- Trend Surface Analysis and Kriging
- Incorporates concept of randomness
- Interpolated surface is conceptualized as one of many that might have been observed, all of which could have produced known data points
- Allow for statistical significance of surface and uncertainty of predictions to be calculated
Gradual Interpolators
- Surface is smooth, no sharp boundaries
- Gradual changes
What can the statistical significance and uncertainty of a surface aid with?
- Use errors to ID places that need more sampling (margins, edge effects)
Abrupt Interpolators
- Quickly changing but continuous values
- Impermeable barriers (eg geological faults)
- High contrast from one variable to the next
Deterministic Methods
- Do no use probability theory
Distance Weighted
- IDW
- Simple
- Assigns values using weighted average
- Local algorithm, decide over what distance to process algorithm
IDW problems
- No assessment of prediction errors
- Can produce ‘bulls eyes’ around data locations
How does IDW work?
- Weights are a decreasing function of distance
- w (d) = 1/d^P, where p is power
- Larger the P, the greatest influence to values closest to the interpolated point
- Weights values higher that are close by to create predicted value
Would additional points improve IDW?
- Probably
- But if variable does not change much over an area then additional points can be redundant
- Possibly costly to collect redundant points
TSA, Fitting Polynomials: Global vs. Local
- Global: Single surface function for entire map
- Local: Estimate surface using only a selection of nearest points
TSA: Multiple functions
- Using a continuous to produce a surface, y = f (x1, x2…xn)
- 1 dependent variable, y
- k independent explanatory variables, x1, x2…xn
TSA: Power Functions
y = f(x^n)
- Linear: y = f (x) = a plus bx, no bend
- Quadratic: y = f (x^2), one bend
- Cubic: y = f (x^3), 2 bends
TSA: Fitting Polynomials
- Decomposes each observation on a spatially distributed variable into a component associated with global trends and a component associated with purely local effects
- Looks at residuals to see difference from global data
- De-trends data
De-trending data
- Fit a polynomial function to find underlying trend
- See if residuals are different from global trend
- Separate and pull out trend and analyze data w/out it
- Can put trend back in later
TSA: What is the trend?
- Regression line
- What can help predict y
- Subtract trend from point to get residual
TSA: Polynomial eqns
y = alpha n x x^n plus alpha n-1 x x^n-1 plus…alpha 0 x x^0
TSA: Linear trend
- Uses a polynomial regression to fit a least-squares surface to input points
TSA: 1st order linear trend surface interpolation
- Performs a least-squares fit of a plane to the set of input points
TSA: Flat plane
- No bend
- Is a 1st order polynomial
- Linear
- Tilted plane
TSA: 2nd degree polynomial
- 1 bend
- Quadratic
- A hill or a valley
TSA: 3rd degree polynomial
- 2 bends
- Cubic
- Hills and valleys
What technique in particular uses multiple regression?
- TSA
- Explanatory variables are x,y coordinates, sometimes completed by higher order polynomials
TSA: Method
- Observed value (zi) = Trend component [f (xi,yi)] plus Local Residual (ei)
- Ordinary Least Squares (OLS)
- Calculate parameters of f while ensuring minimum error
TSA: Levels of analysis
- Description (possible prediction of unknown values
- Remove spatial form (for further analysis)
- Spatial form (model to test hypothesis about spatial variation)
TSA: Series of?
- x, Longitude/Easting
- y, Latitude/Northing
- z, Height or attribute value
TSA: What is the observed phenomenon treated as?
A surface overlaying the map (Cartesian plane, coordinates may be arbitrary)
TSA: What are the 2 scales of spatial processes
- Regional Trend, or systematic component
- Local Effects, or departures from trend
Least Squared Analysis: What is needed prior to analysis?
- Unknown parameters
- Observations
- Mathematical relationship btwn the unknowns and observations
- Then, linearize mathematical relationship (if not already) to transform
- Then apply, OLS formulas
TSA: Original surface
= Trend plus Deviations
TSA: Trend (regional)
Explained component of model
TSA: Deviations (local)
Unexplained component of model
- Residuals
When does original surface = trend?
When error is 0
- Original surface and estimated trend should have same mean value
TSA: Significance Test
- Used to find if polynomial order used is significant
- [R^2K/(K-1)]/[(1-R^2K)/(n-K)]
- Eqn is basically Sum Squares/k-1 or Mean Sum Squares = Sum square/Dof
TSA: Improvement Test
- Used to find if the polynomial is significantly better than the polynomial one degree lower
- [(R^2K-R^2k)/(K-k)]/[(1-R^2K)/(n-K)]
- Eqn is basically (Sum Square K - Sum Square k)/ (K-1) - (k-1)
SSR < 4, R^2 <0.2
Slight, almost negligible trend
SSR 4 - 16, R^2 0.2-0.4
Low, definite, but small trend
SSR 16-49, R^2 0.4-0.7
Moderate-substantial trend
SSR 50-80, R^2 0.7-0.9
Huge, marked trend
SSR 80-100, R^2 0.9-1.0
Very marked trend
SSR
Sum of Squares of Residuals
- Explained
- Residuals = Unexplained
F0 = ?
Mean squares/Mean square error
- Test statistic
TSA: Problems
- Simplicity of polynomial surface compared to natural surface
- 1st is a dipping flat plane, 2nd may have only 1 max or min
- trend surface cannot pass through all data points, it is more of an average - Polynomial surfaces tend to accelerate w/o limit to higher or lower values in areas with no control points
- Seriously exaggerated estimates
TSA: Problems, What about increasing the order of the polynomial?
- Computation difficulties
- Requires sol’n to large number of simultaneous eons whose elements may consist of extremely large numbers
- Maxtrix sol’n may become unstable, or round errors may result, leads to erroneous TS coefficients
TSA: Practical Problems
- Few data points = extreme values seriously distorting surface
- Surfaces susceptible to edge effects, higher order poly’s can turn abruptly near edges leading to unrealistic values
- TS are inexact interpolations, long-range models, extreme values of distant data points can exert unduly large influence, resulting in poor local estimates of studied variable
TSA: big K vs. little k in eqn’s
- k’s are for number of variables present in polynomial order
- K = larger order polynomial, 2nd order has 5 (x,y,xy,x2,y2)
- k = smaller order, 1st has 2 (x,y)
TSA: What is the general process for determining which order is better?
- test significance of 1st and 2nd order
- If 1st is significant but 2nd isn’t, then choose 1st
- If both are significant, then test improvement
- If 2nd is improved and significant but has redundant variables, then take out those and run again and test if still better
- If 2nd better than first, then repeat to compare 2nd w/ 3rd
