Spatial Sampling & Interpolation

Summary

The video script delves into the intricacies of spatial sampling, a critical process in gathering and analyzing geographical data. It explains the concept of a sample frame and the importance of unbiased sampling where every element has an equal chance of selection. However, due to the complexities of spatial data, biased sampling is sometimes necessary, especially when proximity influences relationships among objects. The script outlines various sampling methods, including random, systematic, stratified, cluster, and transect sampling, each with its advantages and limitations. It also touches on the challenges of implementing random sampling in practice and the trade-offs between representativeness and resource investment. Furthermore, the script introduces spatial interpolation as a technique to estimate values between sampled points, leveraging the spatial autocorrelation found in continuous fields. This method is particularly useful for estimating weather conditions, elevation, and adjusting raster image resolutions. The inverse distance weighting example illustrates how spatial interpolation can be applied effectively.

Takeaways

• 📐 **Spatial Sampling Definition**: Selecting points from within an area, known as the sample frame, to gather data.
• ⚖️ **Bias in Sampling**: A sample is biased if elements have unequal chances of selection, while unbiased sampling gives every element an equal chance.
• 🌐 **Spatial Autocorrelation**: Objects close to each other are more likely to be related, which is important in spatial sampling.
• 🔢 **Random Sampling**: A straightforward method where x and y coordinates are randomly generated within the sample frame.
• 🔄 **Repeated Sampling**: Taking multiple random samples to increase representativeness, but it's more time and resource-intensive.
• 📏 **Systematic Sampling**: Imposing a regular grid on the sample space to ensure evenness, but it may miss features with regular patterns.
• 🤝 **Combined Sampling**: Using a combination of systematic and random sampling to balance structure and randomness.
• 🏪 **Cluster Sampling**: Intensively sampling features within selected clusters, useful for focusing on specific areas.
• 🚶 **Transect Sampling**: Focusing sampling efforts on a specific area of interest, efficient but requires good pre-existing understanding.
• 🌳 **Sample Quantity**: The number of samples needed depends on the spatial homogeneity or heterogeneity of the population.
• 📈 **Spatial Interpolation**: Estimating values between known samples for continuous fields, using patterns observed in the dataset.
• 🔍 **Inverse Distance Weighting**: A spatial interpolation technique that estimates values based on distance from known points, with closer samples weighted more.

Q & A

What is spatial sampling?

Spatial sampling is the process of selecting points from within an area, known as the sample frame, to gather spatial data for analysis.

Why is it impossible to capture and describe everything about the world at once?

The world is essentially infinitely complex, making it impossible to capture and describe everything due to the vast amount of information and variables involved.

What are the two main types of sampling that can be applied to spatial data?

The two main types of sampling are unbiased (where each element has an equal chance of being selected) and biased (where some elements have a greater or lower chance of being selected).

Why might we intentionally design our samples to be biased in spatial sampling?

In some cases, such as geography, objects that are close to each other are more likely to be related, so a biased sample that focuses on these relationships can be more useful than a completely random one.

What are the advantages of a random sample?

A random sample is fairly easy to define and implement in theory, providing a good range of possible values in a distribution, and it helps to avoid bias.

What are the potential drawbacks of random sampling?

Random sampling might result in oversampling of large homogeneous areas or missing smaller elements, leading to an unrepresentative sample of the population.

What is systematic sampling and how does it differ from random sampling?

Systematic sampling involves imposing a regular grid on the sample space to ensure evenness. It differs from random sampling by introducing a structured bias, which can make the sample more spatially representative but may miss some features with regular patterns.

How can we combine the benefits of systematic and random sampling?

By defining a regularly spaced grid and then taking random samples within them, we can induce some randomness into the sampling method, which helps to alleviate issues with overlapping periodic features.

What is cluster sampling and when is it beneficial?

Cluster sampling involves intensive sampling of features in clusters around selected locations. It is beneficial when focusing on certain areas, such as shopping centers for a survey, as it allows for efficient targeting of the desired population.

What is transect sampling and how is it used?

Transect sampling allows for focusing efforts on a specific area of interest, making it an efficient way to sample a key feature without sampling outside of the focus area. It is commonly used along linear features like roads or rivers.

How does the number of samples required to represent a population relate to the spatial homogeneity of the area?

The number of samples required is a function of how similar the units of a population are. More homogenous areas require fewer samples, while areas with high spatial heterogeneity need more samples to adequately represent the population.

What is spatial interpolation and why is it used?

Spatial interpolation is a technique used to estimate values of a continuous field at places where measurements are not available. It is used to fill in gaps between samples, making estimations between weather stations, estimating elevation, and changing the resolution of raster images.

How does inverse distance weighting work in spatial interpolation?

Inverse distance weighting is a method where values are interpolated based on their distance from known points, with closer samples given more weight than distant ones. The further away an object is, the less weight it has in the interpolation process.

Outlines

00:00

📐 Understanding Spatial Sampling

The first paragraph introduces the concept of spatial sampling, which is the process of selecting points within an area known as the sample frame. It discusses the complexity of the world and the need to focus on subsets of data. The narrator explains the importance of unbiased sampling where every element has an equal chance of being selected, contrasting it with biased sampling that is sometimes intentionally designed in spatial contexts. The advantages of random sampling are highlighted, including its ease of implementation, but the potential drawbacks, such as oversampling or missing critical elements, are also noted. The paragraph concludes by mentioning the possibility of repeated sampling to improve representativeness, despite its resource-intensive nature.

05:01

🔍 Biased and Systematic Sampling Methods

The second paragraph delves into alternative sampling methods used in geography, focusing on biased and systematic sampling. It contrasts the imperfections of systematic sampling, which may miss regularly patterned features, with the potential adjustments that can be made to compensate. The paragraph explains the concept of a systematic sample, which involves imposing a regular grid on the sample space for evenness, and discusses its suitability for areas with few features and abrupt boundaries. The limitations of systematic sampling for periodic features are also highlighted. The narrator then suggests combining systematic and random sampling to address issues with overlapping periodic features, and introduces stratified random sampling and cluster sampling as additional methods, emphasizing their utility in specific contexts.

10:02

🌳 Efficient Sampling Techniques

The third paragraph discusses efficient sampling techniques like cluster sampling and transect sampling. Cluster sampling involves intensive sampling around selected locations, which is beneficial when focusing on specific areas. Transect sampling allows for focused efforts on an area of interest, which is efficient but requires a good understanding of the spatial structure. The paragraph also addresses the question of how many samples are needed, explaining that it depends on the homogeneity of the population. It emphasizes the importance of knowing the study area to determine the best sampling method, balancing effective coverage with the cost of data collection.

15:03

🔗 Spatial Interpolation for Data Estimation

The fourth paragraph explores spatial interpolation, a technique used to estimate values between sampled points in a dataset. It explains that interpolation is an 'intelligent guesswork' that makes reasonable estimates of continuous fields where measurements are absent. The narrator distinguishes between linear interpolation in one dimension and spatial interpolation in two or three dimensions. The concept of inverse distance weighting is introduced as a method of spatial interpolation that gives more weight to closer samples, aligning with Tobler's First Law of Geography. The paragraph concludes with a mention of the practical applications of spatial interpolation, such as estimating weather conditions between stations, measuring elevation, and changing the resolution of raster images.

Keywords

💡Spatial Sampling

Spatial sampling refers to the process of selecting points within a certain area, known as the sample frame, to collect data. It is a crucial part of gathering spatial data and is integral to the video's theme of understanding how to effectively collect and analyze geographic information. The script discusses how the complexity of the world necessitates sampling due to the impossibility of capturing everything at once. An example from the script is the selection of areas within a frame to either focus on a subset of data or to manage overly complex areas.

💡Sample Frame

A sample frame is the defined area from which spatial data is collected through sampling. It is a fundamental concept in the video as it sets the boundaries within which the sampling process occurs. The script mentions the sample frame in the context of selecting areas for data collection, emphasizing that not all areas within the frame are selected, but rather a subset is chosen based on the study's requirements.

💡Biased Sample

A biased sample occurs when some elements within the sample frame have a greater or lower chance of being selected than others. This concept is significant in the video's discussion on scientific sampling, where the goal is often to achieve an unbiased sample. However, the video also notes that in spatial sampling, sometimes a biased sample is intentionally designed to focus on specific areas of interest. An example given is that if objects close to each other are more likely to be related, a biased sample might be more appropriate.

💡Unbiased Sample

An unbiased sample is one where every element of interest within the sample frame has an equal chance of being selected. This ensures that the sample is representative of the entire population. The video contrasts this with a biased sample and discusses the theoretical ease of implementing an unbiased sample through random selection. However, it also points out the practical challenges, such as the possibility of missing certain elements or oversampling homogeneous areas.

💡Random Sampling

Random sampling is a method where sample points are selected randomly within the sample frame, giving each location an equal chance of being chosen. The video explains that while this method is straightforward in theory, it can lead to issues like oversampling or missing smaller elements. An example from the script is the potential for random samples to miss certain tree species if they are not adequately represented within the randomly selected sample points.

💡Systematic Sampling

Systematic sampling is an approach where a structured pattern, such as a grid, is imposed on the sample space to ensure evenness in the selection of sample points. This method is highlighted in the video as a common practice in geography, as it can be more efficient and spatially representative than random sampling. However, the video also notes its limitations, such as the potential to miss features with regular patterns or clusters.

💡Stratified Random Sampling

Stratified random sampling involves dividing the population into different strata or layers and then taking a random sample from each stratum. The video discusses this as an alternative approach to sampling that can be more representative of the entire population. It is particularly useful when there are distinct subgroups within the population that need to be separately analyzed. An example provided is using grid methods with random intervals to ensure a more balanced representation.

💡Cluster Sampling

Cluster sampling is a technique where intense sampling is conducted within selected clusters or groups of features. This method is beneficial when the focus is on specific areas or features that are naturally clustered. The video uses the example of conducting a survey of shoppers' opinions within shopping centers rather than in less relevant areas like city parks, highlighting the efficiency and relevance of cluster sampling in such contexts.

💡Transect Sampling

Transect sampling involves focusing sampling efforts along a linear feature, such as a road or river. This method is efficient for studying features that are aligned in a linear pattern. The video emphasizes that it requires a good understanding of the spatial structure of the area being studied. An example given is defining transects across a stream to measure its profile, which is more manageable than random sampling across the entire area.

💡Spatial Interpolation

Spatial interpolation is a technique used to estimate values of a continuous field at locations where direct measurements are not available. It is a key concept in the video as it allows for the estimation of data points between known samples. The video explains that this method is particularly useful for fields like temperature, precipitation, and elevation, where there is a strong spatial autocorrelation. An example used is estimating the average temperature across Canada by interpolating between values at different weather stations.

💡Inverse Distance Weighting

Inverse Distance Weighting (IDW) is a spatial interpolation method where values are estimated based on their distance from known points, with closer points having more influence on the interpolated value. This concept is introduced in the video as a way to fill in gaps between sample points, leveraging the principle that closer samples are more similar. The video provides a visual example of how IDW can be used to interpolate elevation data across space, given a few sample points.

Highlights

Spatial sampling involves selecting points from within an area, known as the sample frame, to gather data.

The world is infinitely complex, necessitating the collection of a subset of information or samples to understand it.

Scientific sampling requires each element in the sample frame to have a pre-specified chance of selection to avoid bias.

In some cases, spatial sampling is intentionally designed to be biased to account for the spatial relationships between objects.

Random sampling is theoretically easy to define but can lead to oversampling of large homogeneous areas or missing smaller elements.

Repeated random sampling can lead to a more representative sample but is more time-consuming and resource-intensive.

Systematic sampling involves imposing a regular grid on the sample space to ensure evenness, but may miss features with a regular pattern.

Combining systematic and random sampling can help alleviate issues with overlapping periodic features.

Stratified random sampling involves spacing grids at random intervals to maintain some randomness in the sampling method.

Cluster sampling focuses on intensive sampling of features in clusters around selected locations, useful for targeting specific areas.

Transect sampling allows for focused efforts on specific areas of interest, efficient for linear features like roads or rivers.

The number of samples required depends on the spatial homogeneity of the population; more heterogeneity requires more samples.

Knowledge of the study area is crucial for determining the best sampling method to balance effective coverage with cost.

Spatial interpolation is a technique used to estimate values between known samples, particularly useful for continuous fields.

Inverse distance weighting is a spatial interpolation method that estimates values based on distance from known points.

Spatial interpolation exploits Tobler's First Law of Geography, giving more weight to closer samples.

Continuous fields like temperature, precipitation, and elevation exhibit strong spatial autocorrelation, making spatial interpolation effective.

Spatial interpolation is used for estimating weather conditions between stations, measuring elevation, and changing raster image resolutions.

The choice of sampling method depends on the features being studied and the resources available for data collection and analysis.

Transcripts

all right now i'm going to talk a bit

about spatial sampling

so how do we collect and gather spatial

data

and then um i'll loop back around to

talking a little bit about how we can

exploit a little bit of the special

nature of spatial data

you can think of sampling as the process

of

selecting points from within an area

this area

is also sometimes referred to as the

sample frame

we select some areas from within the

frame but we discard

other right so as i've talked about a

bit already

the world is essentially infinitely

complex there's no way

that we can capture and describe

everything about it all at once so

depending on

the task at hand maybe we focus on just

one subset and collect all the

information we can about it

or maybe the reality or the

area that we're looking at is too

complex and we have to just

stick to looking at a subset so we take

samples

of that area and use that to work with

but how do we choose which points we can

keep and determine

the quality of our data

scientific sampling requires that each

element in a sample frame has

some pre-specified chance of selection

if some elements have either a greater

or lower

chance of being selected then our sample

is said to be biased

and if every element of interest has an

equal chance of being selected then our

sample is said to be

unbiased now biased

isn't necessarily a bad thing in the

context of spatial sampling sometimes we

explicitly design our samples to be

biased

even though again in many of the

hard sciences you want an unbiased

sample

sometimes that's just not a feasible

option in geography

because again objects that are close to

each other are more likely to be related

things that are farther away are more

likely to be

less related in theory

a completely random that is unbiased

sampling process is best with this

each location has an equal chance of

being selected

one of the advantages of a random sample

is that it's

fairly easy to do at least in theory

i'll explain why it's not always

easy in practice but it's easy to define

you create your sample frame that's the

area that you're looking at

and then you just randomly generate x

and y coordinates

within the sample thing typically

speaking this

provides a good range of possible values

in a distribution

however sorry the dog is chewing on a

squeaky toy

however there is a chance that all

samples will be taken from the same type

of elements within a certain population

so for instance we might only sample

from urban areas

and miss forested areas and it's often

difficult to implement this

in practice so if you look at the figure

down here

you can see some of the drawbacks of

random sampling

so all the blue circles are the samples

we take

and with random samples you might end up

by chance

over sampling large elements like a

building or

crop field so these are homogeneous

areas where you don't need a lot of

samples to describe what they are

or you might completely miss smaller

elements so

for instance maybe there's some certain

tree species in an area that you

need to get samples from

if you rely on a random sample maybe

you'll miss all those trees

and then you won't include that in your

abstraction of reality

we have another we have a number of

approaches to limit

the impact of this we can do repeated

sampling

so theoretically if you repeat a random

sample over and over and over again

you'll get a lot more samples and it'll

be more representative of the entire

population

the downside to this is that it's more

time consuming and it requires more

resources

typically speaking collecting data

physically

going out and doing it is fairly

expensive

um you have to pay people to collect the

data or

invest your own time in it it's labor

intensive it's a lot of work

if you've ever done any field work you

know that it's

[Music]

a lot so typically we don't want to do

just random samples you can end up with

a lot of redundancy of labor and things

like that

generally speaking in geography we rely

on biased

systematic sampling methods instead so

this is where we create

some sort of sampling design that trades

a sampling scheme

for randomness so instead of complete

randomness we define some structure

we induce some bias to our sample but we

do that with the intent

of making our sample easier more

spatially representative things like

that

so systematic sampling isn't perfect

either it may miss some features that

have a regular pattern or

cluster but we can make some adjustments

to that

to compensate

a systematic sample is just where some

sort of regular grid is

imposed on the sample space to ensure

evenness so this can be a solid strategy

for areas that contain

only a handful of features with abrupt

boundaries like buildings or something

but it's not ideal for things that

exhibit some degree of periodicity

that is like regular intervals like rows

because you might end up over sampling

or under sampling

the feature if all of your sample points

match

up with the period of whatever

entity you're looking at so for instance

if you

overlay a grid on a road network

it's possible that all of these sample

points might completely miss

the road network it's also possible that

the

samples might mostly fall on the roads

so the purely systematic sample

generally speaking isn't the best option

here

one workaround is you can combine

the benefits of the systematic sample

with the benefits of a random sample

generally speaking the systematic sample

is nice because

it makes your sampling scheme a bit more

organized and regular it's usually

easier to collect

but you get rid of all the randomness

instead you can address this issue

by defining a regularly spaced grid and

then taking

random samples within them so this

induces some randomness

to the sampling method and will

alleviate issues

with overlapping periodic features

however this goes back to the main issue

with

one of the main issues with random

sampling is that it's

time consuming and costly

alternatively another form of stratified

random sampling is you can

rely on the grid method as in your d on

the bottom left

but you can space the grids

at random intervals instead it's

essentially doing the same thing

another option is cluster sampling

so you can do intense sampling of

features in

clusters around a number of selected

locations

this is useful if you know that you want

to focus

on certain areas so for instance

shopping centers

maybe you want to do a survey of

the opinions of shoppers for some

certain thing

right it might make sense to send

canvassers to specific shopping centers

to interview people who pass by

it wouldn't make sense to send one of

your canvassers to a city park to ask

them their opinion on

something if it relates to shopping it

wouldn't be a good idea

to have the surveyors go door-to-door

right you're not necessarily going to be

getting the

target population you're looking for

that would be a waste of resources to go

to a park

or to canada's neighborhood when they

could simply stand outside of the

shopping center and get the target

population they're looking for

so that's one context where cluster

sampling is very beneficial if you

are working with things that are

specifically clustered and you know you

just want to target those things you can

ignore the rest of the sampling area

another option is to just randomly

define

the or you can select the cluster

locations at random

and then intensively sample those

clusters

so this could be beneficial if you're

trying to do a vegetation survey for

instance

you have a large area and you randomly

define

vegetation plots and then you take

samples within those vegetation plots

so again this will this allows you to

use some of the randomness and exploit

some of the advantages

of unbiased sampling while also

exploiting some of the

time saving advantages of biosampling so

you choose a handful of random locations

and then intensively sample them

rather than sporadically sampling a

whole bunch of random

locations this is really efficient time

wise but it might

might not be representative of the

broader population that you're

interested in

another option is transect sampling this

allows you

to focus your efforts only on a specific

area of interest

it's really efficient way to sample just

the key feature that you're looking at

you don't sample outside of the focus

area

but one drawback

is that it requires really good

pre-existing understanding

of the spatial structure of the object

you

or whatever you're studying for maximum

effectiveness

the most common application for this

type of sampling method might be

along linear features like roads or

rivers

so you want to get a stream profile you

just

define transects across the stream

and so you just focus on those specific

areas you don't need to focus

outside of the stream and

uh also because of the

sort of effort and difficulty involved

with like

measuring stream depth you have to have

somebody walk across that stream it

might be easier to just have them walk

in lines across the river rather than

trying to navigate it and

doing random samples at a bunch of

different locations

and then within the transect sample

samples can either be randomly as

randomly spaced stratified or some

combination of the two approaches

so with most of these different types of

sampling methods you can induce some

degree of randomness to make them a

little bit

less biased if you want

how many samples do we need the number

of samples

required to adequately represent a

population is a function

is a function of how similar the units

of a population are

so if you think about the spatially

homogeneous tree farm example

if we have a field where all of the

trees are the same species

and they were planted in the same year

they're all given the same amount of

fertilizer grown under the same light

conditions

there's going to be some small degree of

variability

in tree height so you want to measure

the average height of the trees across

this field

you might need to sample say 30 trees

and go around and measure their height

you can take the average and get a good

idea

of the average tree height on this tree

farm because

it's fairly homogenous they're all

essentially the same

whereas if you want to measure the

average tree height in a natural

landscape with a high degree of spatial

heterogeneity

you're going to need more samples you've

got different tree species

the trees all sprout in different years

especially if it's like an old growth

forest

you've got different soil conditions a

variety of factors that are going to

lead to trees of different heights

spread across the landscape so you're

going to need a lot more samples to

determine the average tree height in

this landscape

than this landscape and

due to spatial heterogeneity because

structure can vary widely across the

landscape

it's important to have a bit of

knowledge about your study area this is

going to help you determine what the

best

sampling method is because the goal is

to maximize returns for minimal effort

need to balance effective coverage with

the cost of collecting the data

and i mean cost not just in money but in

time and

in effort so as is often going to be the

case in this course there's no one

right answer the specific type of

sampling method that you're going to use

will oftentimes depend on the features

that you're looking at and the resources

you have

to conduct whatever analysis you're

working on

so i want to now i'm going to

briefly take a moment to talk about one

of the techniques that we can use

to exploit the special nature of spatial

data

what if we have a data set where we took

some samples

but we want to know what falls between

the samples

we can do something called spatial

interpolation to figure out

what might go this is a really simple

one-dimensional example so you've got a

string of numbers one

two four and five we didn't sample the

middle one

we want to know what's there i think we

can all

assume that it's going to be a three

if you take that to two dimensions

things get a little bit more complicated

but again you can fill in the blanks

using the patterns observed in the data

set

so this is known as spatial

interpolation

and the process of or

the process of filling in blanks in

general is known as interpolation so if

you do it over

one dimension one two three four five

that's known as linear interpolation but

if you do it over two or three

dimensions then we're going to call it

spatial interpolation essentially it's

just

intelligent guesswork where we attempt

to make a reasonable estimate of

values of a continuous field at a place

where we don't have measurements

spatial interpolation only makes sense

for a

continuous field so something

where it varies across space like

temperature or precipitation or

elevation

with categorical data like land cover

it can be a pretty problematic uh thing

to do and it doesn't work very well

the spatial interpolation works really

well with rainfall

temperature pressure most weather

observations in general

so you can use it to make estimations

between

weather stations so canada

has a network of weather stations spread

across the country

some more dense and populated areas some

more sparse especially in the north

but we can use the information from

these weather stations to

estimate the average temperature at all

spaces

at all locations across the country by

interpolating between the values at

weather stations

it can be used to measure or estimate

elevation between measured locations

and it's also used when we change

resolution of raster images

so all spatial interpolation methods

incorporate distance to known samples

and if this sounds familiar that's

because they exploit tobler's first law

of geography

closer samples are going to be given

more weight than

distant ones and a threshold is usually

set to determine the maximum distance to

take samples from

most continuous fields tend to exhibit

very strong

spatial autocorrelation so it's pretty

reasonable to assume

that values that are missing are likely

to be similar

to those that are around them in the

field

so here's just a visual example of how

you might go about doing spatial

interpolation

with elevation data there is a tool

called

inverse distance weighting where

values are interpolated

based on their distance from known

points

where the farther away

the object is the less weight it has

that is it is inversely weighted with

its distance

so you can see a one-dimensional and

two-dimensional example

of inverse distance weighting where on

the left

using it to just interpolate the

values between points on a line and on

the right we're using it to

estimate elevation across space given

a small number of samples

so i will discuss or i'll show an

example

later on in the term of specifically how

to do inverse distance weighting

in arcgis pro but i just wanted to

introduce this concept because it kind

of ties in nicely with what i've already

talked about