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Geog418SptStts > Spatial Interpolation & Trend Surface Analysis > Flashcards

Spatial Interpolation & Trend Surface Analysis Flashcards

1
Q

Spatial interpolation background

A
  • 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
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2
Q

What can spatial interpolation be used for in GIS?

A
  • 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)
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3
Q

Spatial interpolation is often used to aid what?

A
  • Spatial decision making process both in physical and human geography
  • ex. Mineral prospecting and hydrocarbon exploration
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4
Q

4 Types of Spatial Interpolation

A
  • Global/Local
  • Exact/Approximate
  • Stochastic/Deterministic
  • Gradual/Abrupt
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5
Q

Global Interpolators

A
  • Determine a single function which is mapped across the whole region
  • A change in one input value affects the entire map
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6
Q

Local Interpolators

A
  • Algorithm is applied repeatedly to a small portion of total points
  • A change in one input value only affects the result within the window
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7
Q

Exact Interpolators

A
  • Surface passes through all points whose values are known
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8
Q

Approximate Interpolators

A
  • Used when there is some uncertainty about the given surface values
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9
Q

Stochastic Method

A
  • 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
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10
Q

Gradual Interpolators

A
  • Surface is smooth, no sharp boundaries

- Gradual changes

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11
Q

What can the statistical significance and uncertainty of a surface aid with?

A
  • Use errors to ID places that need more sampling (margins, edge effects)
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12
Q

Abrupt Interpolators

A
  • Quickly changing but continuous values
  • Impermeable barriers (eg geological faults)
  • High contrast from one variable to the next
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13
Q

Deterministic Methods

A
  • Do no use probability theory
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14
Q

Distance Weighted

A
  • IDW
  • Simple
  • Assigns values using weighted average
  • Local algorithm, decide over what distance to process algorithm
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15
Q

IDW problems

A
  • No assessment of prediction errors

- Can produce ‘bulls eyes’ around data locations

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16
Q

How does IDW work?

A
  • 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
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17
Q

Would additional points improve IDW?

A
  • Probably
  • But if variable does not change much over an area then additional points can be redundant
  • Possibly costly to collect redundant points
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18
Q

TSA, Fitting Polynomials: Global vs. Local

A
  • Global: Single surface function for entire map

- Local: Estimate surface using only a selection of nearest points

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19
Q

TSA: Multiple functions

A
  • Using a continuous to produce a surface, y = f (x1, x2…xn)
  • 1 dependent variable, y
  • k independent explanatory variables, x1, x2…xn
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20
Q

TSA: Power Functions

A

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
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21
Q

TSA: Fitting Polynomials

A
  • 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
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22
Q

De-trending data

A
  • 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
23
Q

TSA: What is the trend?

A
  • Regression line
  • What can help predict y
  • Subtract trend from point to get residual
24
Q

TSA: Polynomial eqns

A

y = alpha n x x^n plus alpha n-1 x x^n-1 plus…alpha 0 x x^0

25
Q

TSA: Linear trend

A
  • Uses a polynomial regression to fit a least-squares surface to input points
26
Q

TSA: 1st order linear trend surface interpolation

A
  • Performs a least-squares fit of a plane to the set of input points
27
Q

TSA: Flat plane

A
  • No bend
  • Is a 1st order polynomial
  • Linear
  • Tilted plane
28
Q

TSA: 2nd degree polynomial

A
  • 1 bend
  • Quadratic
  • A hill or a valley
29
Q

TSA: 3rd degree polynomial

A
  • 2 bends
  • Cubic
  • Hills and valleys
30
Q

What technique in particular uses multiple regression?

A
  • TSA

- Explanatory variables are x,y coordinates, sometimes completed by higher order polynomials

31
Q

TSA: Method

A
  • Observed value (zi) = Trend component [f (xi,yi)] plus Local Residual (ei)
  • Ordinary Least Squares (OLS)
  • Calculate parameters of f while ensuring minimum error
32
Q

TSA: Levels of analysis

A
  • Description (possible prediction of unknown values
  • Remove spatial form (for further analysis)
  • Spatial form (model to test hypothesis about spatial variation)
33
Q

TSA: Series of?

A
  • x, Longitude/Easting
  • y, Latitude/Northing
  • z, Height or attribute value
34
Q

TSA: What is the observed phenomenon treated as?

A

A surface overlaying the map (Cartesian plane, coordinates may be arbitrary)

35
Q

TSA: What are the 2 scales of spatial processes

A
  • Regional Trend, or systematic component

- Local Effects, or departures from trend

36
Q

Least Squared Analysis: What is needed prior to analysis?

A
  • Unknown parameters
  • Observations
  • Mathematical relationship btwn the unknowns and observations
  • Then, linearize mathematical relationship (if not already) to transform
  • Then apply, OLS formulas
37
Q

TSA: Original surface

A

= Trend plus Deviations

38
Q

TSA: Trend (regional)

A

Explained component of model

39
Q

TSA: Deviations (local)

A

Unexplained component of model

- Residuals

40
Q

When does original surface = trend?

A

When error is 0

- Original surface and estimated trend should have same mean value

41
Q

TSA: Significance Test

A
  • 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
42
Q

TSA: Improvement Test

A
  • 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)
43
Q

SSR < 4, R^2 <0.2

A

Slight, almost negligible trend

44
Q

SSR 4 - 16, R^2 0.2-0.4

A

Low, definite, but small trend

45
Q

SSR 16-49, R^2 0.4-0.7

A

Moderate-substantial trend

46
Q

SSR 50-80, R^2 0.7-0.9

A

Huge, marked trend

47
Q

SSR 80-100, R^2 0.9-1.0

A

Very marked trend

48
Q

SSR

A

Sum of Squares of Residuals

  • Explained
  • Residuals = Unexplained
49
Q

F0 = ?

A

Mean squares/Mean square error

- Test statistic

50
Q

TSA: Problems

A
  1. Simplicity of polynomial surface compared to natural surface
  2. 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
  3. Polynomial surfaces tend to accelerate w/o limit to higher or lower values in areas with no control points
    - Seriously exaggerated estimates
51
Q

TSA: Problems, What about increasing the order of the polynomial?

A
  • 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
52
Q

TSA: Practical Problems

A
  • 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
53
Q

TSA: big K vs. little k in eqn’s

A
  • 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)
54
Q

TSA: What is the general process for determining which order is better?

A
  • 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

Geog418SptStts (6 decks)

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