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```Agronomic
Spatial Variability
and Resolution
What is it?
How do we describe it?
What does it imply for
precision
management?
Agronomic Variability
• Fundamental assumption of
precision farming
• Agronomic factors vary spatially
within a field
• If these factors can be measured
then crop yield and/or net economic
returns can be optimize
Agronomic Variables
• Soils
–
–
–
–
Classification
Texture
Organic matter
Water holding
capacity
• Topography
– Slope
– Aspect
• Fertility
–
–
–
–
–
pH
Nitrogen
Phosphorus
Potassium
Other nutrients
• Plant available
water
• Crop Cultivar
Agronomic Variables
• Temperature
• Rainfall
• Weeds
– Species
– Population
• Insects
– Species
– Feeding patterns
• Tillage Practices
• Soil Compaction
• Diseases
– Macro and micro
environment
• Crop Stand
• Method and
Uniformity of
Application
– Fertilizers
– Crop protectants
What is variability
• Variability - difference in the
magnitude of measurements of a
variable
– Values can change randomly because
of error in the sensor
– Values can change because of changes
in the underlying factor
• As time changes (Temporal)
• As location changes (Spatial)
Why statistically describe
measurements?
• Raw data sets are too large to
understand or interpret
• Statistics provide a means of
summarizing data and can be readily
interpreted for making management
decisions
• Statistics can define relationships
among variables
Statistical Analyses Commonly
Used In Precision Agriculture
• Descriptive Statistics
– Measures of Central Tendency
• Mean
• Median
– Measures of Dispersion
• Range
• Standard Deviation
• Coefficient of Variation
• Regression
• Geostatistics - Semivariance Analysis
Measures of Central Tendency
• When a factor, such as crop yield, is
measured at different locations
within a field, values may vary
greatly
• This variation can appear to be
random
• The set of these measurements is a
population
• A value exists that is the central or
usual value of the population
Measures of Central Tendency
• This is important because the current
practices in precision farming treat
the field or area in the field based on
the average of the measurements
within the field or area
Mean or Average Value
• Most common measure of central
tendency
• Definition:
For n measurements X1,X2,X3,…,Xn
n
X 1 + X 2 +...+ X n
=
X =
n
X
i =1
n
i
Mean or Average
• The mean or average value is useful
if the measured value is normally
distributed (Bell Curve)
– Most biological processes are normally
distributed
– Spatially distributed measurements are
often not normally distributed
• To calculated the mean in Excel
= Average (Col Row:Col Row)
Definition of (Col Row : Col Row)
(Col Row:Col Row)
• Column letter of the upper left
cell of an array of data
• Row number of the upper left
cell of an array of data
• Column letter of the lower right
cell of an array of data
• Row number of the lower right
cell of an array of data
• The “:” instructs Excel to
include all data between the two
corner cells
The Median Value
• For skewed distributions, is the
better predictor of the expected or
central value
• Calculated by ranking the values
from high to low
– For an odd number of measurements,
the median is middle value
– For an even number of measurements,
the median is average of the two middle
values
The Median Value
• In Excel, the median is calculated
using the following formula:
= Median (Col Row:Col Row)
Normal vs. Skewed
Distribution
Mean
Skewed
Normal
Skewed
Median
Normal
Skewed
Normal
Measures of Dispersion
• Measures of dispersion describe the
distribution of the set of
measurements
Maximum and Minimum Values
• The maximum value is the highest
value in the data set
• In Excel the maximum value is
calculated by:
= Max(Col Row:Col Row)
• The minimum value is the lowest
value in the data set and is
calculated by:
= Min(Col Row:Col Row)
Range of the Sample Set
• Difference between the maximum
and minimum values of the
measurement
• Calculated in Excel by the following
formula:
= Max (Col Row:Col Row)
- Min (Col Row:Col Row)
Standard Deviation
• The standard deviation of a normally
distributed sample set is 1/2 of the
“range” of ≈68 %values for the
population
n
s=
 (X
i =1
i
-X )
n -1
2
Standard Deviation
• For a normal distribution (Bell Curve)
≈ 95% of the samples from a population
will lie in the interval
X - 1.96s  Z  X + 1.96s
– Where: X is the mean(average) value
Z is a value (measurement)
s is the standard deviation
• The standard deviation is calculated in
Excel using the following formula:
= Stdev (Col Row:Col Row)
Coefficient of Variation
• The magnitudes of the differences
between large values and their means
tend to be large. The differences between
small values and their means tend to be
small.
• Consequently, a high yielding field is
likely to have a higher standard deviation
than a low yielding field, even if the
variability is lower in the high yield field.
Coefficient of Variation
• Thus, variation about two means of
different magnitudes cannot easily
be compared.
• Comparisons can be made by
calculating the relative variation, or
the normalized standard deviation.
• This measurement is called the
Coefficient of Variation.
Coefficient of Variation
• The Coefficient of Variation or C.V. is
calculated by dividing the standard
deviation of the data set by its mean.
Often that value is multiplied by 100
and the C.V. is expressed as a
percentage.
• Experience with similar data sets is
required to determine if the C.V. is
unusually large.
Mean, Standard Deviation and
Coefficient of Variation
Population = Y
Mean Plant
Spacing
CV =
Std. Dev. = s
s
X
Population = ½ Y
Mean Plant
= 2X
Spacing
2 (X - X )
2
Std. Dev. =
n -1
2
CV =
2s s
=
2X X
Correlation
• One objective of precision farming is
to alter the level of one variable (e.g.
soil nitrate) to change the response
of another variable (e.g. grain yield).
• There are other confounding factors
affecting grain yield, such as weed
competition or pH, which cannot
always be accounted for.
Correlation
• The practitioner still needs to
determine the degree to which the
two variables vary together.
• The correlation coefficient or r is that
measure.
• The correlation coefficient, r, lies
between -1 and 1. Positive values
indicate that X and Y tend to increase
or decrease together.
Correlation
• Values of r near 0 indicate that there is
little or no relationship between the two
variables.
• The coefficient of determination or r2 is
important in precision farming because,
when the samples are collected by
location in the field, it indicates the
percentage of the variability in the
dependent variable (e.g. yield) explained
by the independent variable (e.g. N
fertilizer).
Correlation
• For example, if the r2 of soil N and grain
yield is 90% then 90% of the variability
across the field can be explained by soil
nitrate. Spatially varying the N fertilizer
rate based on the nitrate level in the soil
should have a large effect on grain yield.
• In Excel, correlation r is calculate by the
following:
= Correl (Col Row:Col Row, Col Row:
Col Row)
To calculate r2, simply square the value of r.
High Resolution Variability
Study – 1 ft x 1ft Experiments
```
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