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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<title>A Comparison of Spatial Interpolation methods for Air pollution</title>
<meta charset="utf-8" />
<meta name="author" content="Hyesop Shin" />
<<<<<<< HEAD
<meta name="date" content="2019-05-06" />
=======
<meta name="date" content="2019-04-03" />
>>>>>>> 6d9f80ec12b9543a70225f34131deb677ec6b39c
<link href="libs/remark-css/default.css" rel="stylesheet" />
<link href="libs/remark-css/default-fonts.css" rel="stylesheet" />
</head>
<body>
<textarea id="source">
class: center, middle, inverse, title-slide
# A Comparison of Spatial Interpolation methods for Air pollution
## Kriging vs GAM
### Hyesop Shin
<<<<<<< HEAD
### 2019-05-06
=======
### 2019-04-03
>>>>>>> 6d9f80ec12b9543a70225f34131deb677ec6b39c
---
class: left, top
# Things to cover
#### Rationale
#### Kriging approach
#### GAM approach
#### Pros and Cons (from lit review)
#### Comparing results
---
class: inverse, center, middle
# Rationale
---
class: left, top
# Rationale
- Traditionally, spatial interpolation has been widely used to estimate pollution levels across surface areas
--
- The advantages were to get a broad sense of pollution estimation and get outputs with less calculation effort
--
- However, the statistical overfitting from variograms had over/under-estimated the outcomes that smoothed out the variance
--
- Nevertheless, it can still be worthwhile to model the results and compare them with the individual sensor data.
---
class: inverse, center, middle
# Method 1: Generalised Additive Models (GAMs)
---
class: left, top
# GAM: Concept
- GAM is a non-parametric extension of GLMs, and used when the modeller has no **a priori** hypothesis to select any parametric response functions (Borokini 2016, Wood 2017)
--
- e.g. linear, logistic, or quadratic <- `whichever model suits`
--
`$$g(E(Y))= α + s_1(x_1) + ... + s_p(x_p),$$`
--
- where Y is the dependent variable of index
- E(Y) denotes the expected value,
- g( ) denotes the link function that links the expected value
- predictor variables x~1~, … x~p~
--
- Each predictor has a relationship with the dependent variable that can be indicated as a scatterplot
---
class: left, top
# GAM: Concept (cont.)
- **Penalised regression splines**: Once the dots are splitted to sections, the linkage between sections called *knots* are matched with polynomial functions (e.g. piecewise linear regression), then connects the untied points with smooth / link functions (e.g. loess, cyclic cubic, P-spline, tensor product) (Wood, 2017)
--
<img src="interpolation-methods_files/piecewise.png" width="60%" style="display: block; margin: auto;" />
--
- Adding functions to knots to predict the link function is what we call an *additive* model.
- Model fitting is based on likelihood e.g. AIC, RMSE
---
class: left, top
# Advantages
- **Flexibility**: GAM is also known as a semi-parametric model as the assumptions are generous, and the relationships between response and predictors can find their best fit (Wood, 2017)
--
- can be compared to OLS regression or GLMs that underlies strict parametric assumptions
- useful when the relationship between the dependent and predictive variables are expected to be complex, and not easily fitted to a linear estimate
--
- **Additivity**: Categorical predictors are also able to be included in the model e.g. landcover types, gender.
--
- **Avoid overfitting**: Regularisation of predictor function helps avoid overfitting
---
class: left, top
# Disadvantages
--
- **Weakness to overfitting**: Since the model has its elasticity of smoothness for model fitting, a modeller has to be conscious in controlling the wiggliness.
--
- Modellers should take into account two things when fitting a nonlinear model: close to data (avoiding underfitting), and not fitting the noise (avoid overfitting).
--
- Possible alternatives are to check whether the knots are too many or small by using `gam.check` or `qqgam` functions in R.
--
- **Slow execution speed**
--
---
class: left, top
# GAM: example plot
- Despite the difficulty of model interpretation and overfitting, it is preferred in spatial and temporal interpolation created with sophisticated smoothing methods.
<div class="figure" style="text-align: center">
<img src="interpolation-methods_files/gam_ex.png" alt="(A) shows black lines of predicted values, red dotted lines for -1 standard error, and green dotted lines for 1 standard error. (B) shows the observed and modelled relationship of residuals from DEM. Using the GAM approach, a series of interpolated PM&lt;sub&gt;10&lt;/sub&gt; maps are shown in (C)." width="60%" />
<p class="caption">(A) shows black lines of predicted values, red dotted lines for -1 standard error, and green dotted lines for 1 standard error. (B) shows the observed and modelled relationship of residuals from DEM. Using the GAM approach, a series of interpolated PM<sub>10</sub> maps are shown in (C).</p>
</div>
---
class: inverse, center, middle
# Method 2: Kriging
---
class: left, top
# Kriging: Concept
--
- Kriging is a geostatistical interpolation method that predicts an estimate for an unmeasured location, given the characterized mean and variance structures.
--
- This method follows Tobler’s First Law of Geography: *Everything is related to everything else, but near things are more related than distant things*
- by calculating the overall mean, distance, and variance of all observations.
--
`$$\widehat{X}(s_{o}) = \sum_{i=1}^{N} \lambda_{i}Z(s_{i})$$`
- Z(s<sub>i</sub>) = the measured value at the `\(i\)`th location
- `\(\lambda\)`<sub>i</sub> = an unknown weight for the measured value at the ith location
- s<sub>0</sub>$ = the prediction location
- *N* = the number of measured values
- The variance feature that represents spatial dependency is assessed by using a variogram
---
class: left, top
# Kriging: Concept(Cont.)
<div class="figure" style="text-align: center">
<img src="interpolation-methods_files/kriging_concept.png" alt="Conceptual process of Kriging measurement" width="100%" />
<p class="caption">Conceptual process of Kriging measurement</p>
</div>
- A: Distance and the half-squared variances of all points. The dots are then filled into imaginary bins, and finally averaged to a single value.
- B: A model of made from the blue dots.
- C: Using three covariance parameters - range, partial sill, and nugget -, C creates a new covariance function. The range is the distance at which a spatial correlation exists. The partial sill and nugget parameters represent spatial and non-spatial variability, respectively.
---
class: left, top
# Kriging types
Once the variogram is confirmed, the interpolation can be modelled under different mathematical approaches.
- **Ordinary kriging** assumes a constant mean over space
- **Universal kriging** includes covariates in a regression framework to represent local variation
- Other approaches: **co-Kriging, block Kriging, simple Kriging**, and a mixture of either Kriging methods.
---
class: left, top
# Advantages
- Statistically proven analysis: **exploratory stat analysis -> semivariogram -> consider radius -> predict results**
- Explicit calculations for Semivariograms, and easy manipulation in model fitting: **choose semivariogram models, adjust range, nugget**
- *R specific* Small memory storage -> Appropriate for big data processes on local machine
---
class: left, top
# Disadvantages
- It does not consider various **built environments (e.g. roads, bridges, buildings, parks)** nor physical environments (e.g. mountains, weather) that cause a massive difference in our real world.
- Hoek (2017): > Kriging outcomes that only takes pollution fields into consideration can underestimate the spatial variation of locations that are further away from the stations.
--
- Max distance for local Kriging: if the max Dist isn't specified the local boundary will be considered as infinity `Maxdist = inf`, which slows down the modelling speed
--
- Randomness of Semivariogram model selection
--
- Smooths *out* the extremes during interpolation: even if the model is fitted, the smoothness of model may end up over/under-fitted, which in turn will have artifact results or have edge or bull's eye effects
---
class: inverse, center, middle
# Outcomes: Spatial Interpolation
---
class: top, left
<<<<<<< HEAD
# Problems still remain...
=======
# Outcome: Semivariogram - Codes
>>>>>>> 6d9f80ec12b9543a70225f34131deb677ec6b39c
```
myList <- list()
for(i in 1:20) {
nam <- paste("Sem.Var", i, sep = "")
myList[[length(myList)+1]] <- assign(nam,
autofitVariogram(no2.winter[[i+2]] ~ 1,
no2.winter))
}
rm(list=ls(pattern="Sem.Var"))
<<<<<<< HEAD
---
class: inverse, top, left
# Problem solving: Applying Additional Road Effects on Spatial Interpolation Outputs
- Assuming that the pollution impacts are higher near roads, how can we apply additional road effects on top of spatial interpolations?
- With a spatially coarse, but temporarilly abundant data, can we do something *cool*?
--
- To overcome these limitations, an alternative appraoch is to measure **12 hour average of background stations and roadside stations**, **compare the ratio**, then **apply the ratio to road layouts**.
=======
semvar <- lapply(myList, function(x) plot(x))
do.call(grid.arrange, semvar[1:4])
```
>>>>>>> 6d9f80ec12b9543a70225f34131deb677ec6b39c
---
class: top, left
# Outcome: Semivariogram- Plots
<img src="interpolation-methods_files/outcome_semivariogram.png" width="70%" style="display: block; margin: auto;" />
---
class: top, left
# Outcome: Kriging
<img src="interpolation-methods_files/outcome_Kriging.png" width="100%" style="display: block; margin: auto;" />
---
class: top, left
# Outcome: GAM
<img src="interpolation-methods_files/outcome_GAM.png" width="100%" style="display: block; margin: auto;" />
---
class: left, top
# Problems still remain...
- This section introduces a new approach of additional road effect that superimposes spatial interpolation outcomes.
--
- Spatial interpolation itself contains variances and biases, particularly when temporal aggregation interval is short and monitoring stations are distant to each other
- e.g. Land Use Regression
--
- Even in the same study area, the signifance of predictor variables would differ to the previous year or it might not be selected
- e.g. Fluid dynamic models
---
class: top, left
# Solution: Applying Additional Road Effects on Spatial Interpolation Outputs
- Assuming that the pollution impacts are higher near roads, how can we apply additional road effects on top of spatial interpolations?
--
- With a spatially coarse, but temporarilly abundant data, can we do something *cool*?
--
- To overcome these limitations, an alternative appraoch is to measure **12 hour average of background stations and roadside stations**, **compare the ratio**, then **apply the ratio to road layouts**.
---
class: inverse, center, middle
# Outcomes: Ratio Plots
---
class: top, left
## NO<sub>2</sub>: Dec.2013 - Feb.2014
<img src="interpolation-methods_files/ratio_no2.png" width="70%" style="display: block; margin: auto;" />
---
class: top, left
## O<sub>3</sub>: Dec.2013 - Feb.2014
<img src="interpolation-methods_files/ratio_o3.png" width="70%" style="display: block; margin: auto;" />
---
class: top, left
## PM<sub>10</sub>: Dec.2013 - Feb.2014
<img src="interpolation-methods_files/ratio_pm10.png" width="70%" style="display: block; margin: auto;" />
---
class: inverse, center, middle
# Add roads
<img src="interpolation-methods_files/road.png" width="100%" style="display: block; margin: auto;" />
---
class: inverse, center, middle
# Outcomes: Final
---
class: top, left
# Outcomes: GAM + Road ratio
<img src="interpolation-methods_files/outcome_GAM_final.png" width="100%" style="display: block; margin: auto;" />
---
class: top, left
# Outcomes: Kriging + Road ratio
<img src="interpolation-methods_files/outcome_Kriging_final.png" width="100%" style="display: block; margin: auto;" />
### Notice any difference?
---
class: inverse, center, middle
# Thank You
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