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 Topography Fertility Plant available water 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 Insects Species Population Insects 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 Systematic error or bias 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 Normal Distributions 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 dimensions representing Biological Material are generally reported as single “expected” values. Examples: http://www.nue.okstate.edu/By_Plant_Variability_Corn.htm

å Mean or Average Value Most common measure of central tendency Definition: For n measurements X1,X2,X3,…,Xn n å X X + X + . . . + X i = 1 2 n = i = 1 X n n

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) 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, it 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 In Excel, the median is calculated using the following formula: = Median (Col Row : Col Row)

Normal vs. Skewed Distribution Mean Median Normal Normal Skewed Skewed Normal Skewed

Normality Biological materials physical measurements are generally normally distributed about the mean. There are several test of normality which will be discussed in your statistics courses. However, three “quick and dirty” tests can be accessed easily from Excel The first is simply comparing the mean and median values. If the values are nearly the same the measurement is likely distributed normally. Excel has function calls to calculate Skewness and Kurtosis. These statistics can be used to test for normality

Normality Kurtosis measures deviation from the mean. A value of ‘0’ indicates that there is no deviation from a normal distribution. A positive value indicates that more values are clustered near the mean or far from it. A negative value means a “flat” top of the curve. = Kurt (Col Row : Col Row)

Normality Skewness is a measure of the tail of the distribution. A positive value indicates that there is an asymetrical tail of the distribution and that it is positive. A negative value indicates that there is a negative tail to the distribution. =Skew (Col Row : Col Row)

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)

å Standard Deviation ( X - X ) s = n - 1 The standard deviation of a normally distributed sample set is 1/2 of the “range” or ≈68 %values for the population n å 2 ( X - X ) i s = i = 1 n - 1

s X Z 96 . 1 + £ - Standard Deviation For a normal distribution (Bell Curve) ≈ 95% of the samples from a population will lie in the interval 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) s X Z 96 . 1 + £ -

Coefficient of Variation The magnitude 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 or the same as the lower yielding 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 Std. Dev. = s CV Population = ½ Y Mean Plant Spacing Std. Dev. CV

Correlation One objective of Biosystems engineering and Agronomy 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 soil pH, which cannot always be accounted for.

Correlation Scientists still need 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. y y x x

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.

Regression Excel has the capability of fitting mathematical models (linear and non-linear curves) to data which relate dependent to independent variables. Regression (curve fitting) can be performed using the Charting GUI in Excel. You can also directly calculate the slope and intercept for a linear model using the commands

Regression = Intercept (Col Row : Col Row) and = Slope (Col Row : Col Row) Regression R2 is a measure in decimal percent of how well the model fits the data. For linear regression, the regression R2 can be directly calculated be squareing the correlation coefficient

Data presentation Always be wary of Data. What is the error What is the scale of the Axis. Is it a fertilizer Trial, was the a 0 check?

5 bushel and $30 increase due to 2 pt of MikesMagic Juice over 2 gal Joes Sauce

Improving Wheat Profits profit increase $110.70 $88.65 What question if any do you ask?

The 3 R’s r correlation coefficient r2 correlation of determination P and K, slope and texture, N and OM Are they correlated at that site r2 correlation of determination N and yield, irrigation and yield, lime and soil pH Independent (controlled) and dependent (result) R2 Regression how well does a model explain the data. Linear, quadratic, Linear plateau

Regression R2

Spatial Interpolation Interpolation: In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. Methods Proximal / Inverse Distance Moving Average/distance weighted. Triangulation Spline Kriging provides a confidence in estimates produced.

Inverse Distance Weighting Inverse Distance Weighting (IDW) is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the known points. The name given to this type of methods was motivated by the weighted average applied since it resorts to the inverse of the distance to each known point ("amount of proximity") when assigning weights.

IDW Known value, distance between and a Power How much could distance influence value of unknown. identify the power that produces the minimum RMSPE root mean square prediction error Shepard's interpolation in 1 dimension, from 4 scattered points and using p=2.

Kriging Kriging is a group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as a function of the geographic location) at an unobserved location from observations of its value at nearby locations. Kriging belongs to the family of linear least squares estimation algorithms Use of variograms.

Kriging Example of one-dimensional data interpolation by kriging, with confidence intervals. Squares indicate the location of the data. The kriging interpolation is in red. The confidence intervals are in green.

In IDW, the weight, ?i, depends solely on the distance to the prediction location. However, in Kriging, the weights are based not only on the distance between the measured points and the prediction location but also on the overall spatial arrangement among the measured points. To use the spatial arrangement in the weights, the spatial autocorrelation must be quantified. Thus, in Ordinary Kriging, the weight, ?i , depends on a fitted model to the measured points, the distance to the prediction location, and the spatial relationships among the measured values around the prediction location.

Impact of Resolution of samples