UAV Bands and Leaf Nutrient Elements
Main Effects with Nitrogen Treatments: One Factor Experiment
with Multiple Levels by One Way ANOVA
Jessica R. Velos1 , Ian B. Lumiano1 , John Mark V. Manhuyod2
Benjamin Z. Mabanta2 and Joey I. Orajay1
1Agricultural Research Department, Del Monte Philippines Inc.
M.Fortich,Bukidnon, Philippines 8705
email: email@example.com, firstname.lastname@example.org, email@example.com
2Plantation Analytics and Geomatics Department, Del Monte Philippines Inc.
M.Fortich,Bukidnon, Philippines 8705
email: firstname.lastname@example.org, email@example.com
ABSTRACT: A study is made on Field No.11 an experimental area in the plantation with the intent of determining the optimal delivery of Nitrogen when it is controlled in seven treatments namely when A=0% B=25% C=50% ,D=75% ,E=100% F=125% and G=150% all in percentage delivery figures as per Plantation Practice which were done in four replicates. After each delivery, two serial processes commence. First, are mission flights using multispectral sensor UAV for acquiring a set of 4-band sensor response from Green, Red, Red Edge and NIR, this is the basis for computing a select set of vegetation indices CIgreen, CIrededge, NDVI, GRVI. Soon after, a chemistry based laboratory measurement is done to measure for Nitrogen, P, K, Mg, Zn, Fe, Mn, Ca from the leaf samples identified and geotagged in the field that were covered from the flight missions. In this scheme only Nitrogen delivery by spray is the controlled variable and is the primary cause while the laboratory results for the leaf nutrients elements and the UAV sensor bands responses are the observations. Because we have a single quantitative factor that of Nitrogen alone, this renders itself to a single factor experiment that produced several measurement of response variation, which leads us to the use of completely randomized design (CRD) and the Analysis of Variance (ANOVA) statistics. Our results show that a log regression best describes the response for nitrogen delivery from median values as one treatment progresses. Furthermore, the leaf nutrient does not necessarily follow the increase in their chemistry elements present with increasing treatments. While the UAV response to Nitrogen are practically not detectable as adirect measurement from the vegetation indices alone specially at those frequencies where the multispectral bands that lie in Red and NIR (NDVI) other than some indication at the Red Edge(CIre) and Green, Red band (GRVI) as significant in the Pearson correlation test and Main Effects.
1 First look at Nitrogen delivery replication data
The provided data by the Agricultural Research Department comes in Excel sheet where the first column is ”Treatments” from A to G as have been described in the previous section. A second column that follows is ”Nitrogen” under the ”Laboratory Leaf Nutrient Analysis Result the next columns are for UAV Bands and followed by applicable Vegetation Indices. A rearrangement to a row wise instead of column wise for these data were done to accommodate them conveniently in a Minitab c worksheet. In the following page in Figure 1 we show the boxplot of all the 7 treatments in 4 replicates from the delivery of Nitrogen while the average measured Nitrogen per replication is shown in Figure 2. In the 3rd replicate it is showing the lowest value down to about 10% from previous and up again by about 5% to the next replication. If we take each and individual Treatment with their replicates and plot them we can see their variations individually as shown in Figure 3. Lines are joined from one mean value to the other shows the change is one replication to other – as we can see even at A=0% nitrogen delivery the plant has already some amount of nitrogen at the onset however, this goes down on each successive replication. It is noticeable at the second replication all the values practically increased by about 5% but declined in the third and even went down on the fourth replication for the highest treatment at G=150% somewhat recovered for the others. To interpret the box plot in the above figures, we take a look at the length of the whisker lines, one above it shows the largest value and the line below the box is the smallest value at the tip. The middle line along the box is the median value (or 50th percentile). The top edge of the box represent 75th percentile (or 3rd quartile) while the bottom edge 25th percentile. Outliar values are is shown with special characterters here ”*” in Replicate III.
2 Can Nitrogen delivery be modeled: Simple Regression based on field data
The 7 treatments consisting of 4 replications on each is statistically plausible since we don’t expect SD or standard deviation to be felt until like 21 data points following the rule of thumb for SPC or Statistical Process Control. So, if we proceed and simply plot out all the mean values from each replication such as shown in Figure 4 below we see that a linear incremental increase in the delivery of nitrogen having R2=0.91 which quite good already. However, if we could have increased in more having a better R-squared value we could explore the possibility of quadratic line fit as shown in Figure 5 with R2=0.92 a small improvement for goodness of fit. We see that an incremental increase in Treatments from A to G does not necessarily result to a directly proportional increase in Nitrogen content even for the same condition and location of the field. Both regression line can calculate the expected Nitrogen level moving forward from highest possible treatment.
The regression lines above we noticed on the 5th treatment at 100% we see that Nitrogen dipped rather than increased even for 25% increase from the prior treatment. Perhaps stopping the delivery at that point will yield a better treatment than increasing them some more. We show in Figure 6 in the following page the result of a cubic line fit results to a 3rd order polynomial line fit with a very high R2=0.97. It does seem to suggest that highest treatment the could have the highest response is at 75% or at Treatment D or could just be optimal at Treatment E or at 100% of PP. We could see on this regression model suggest that Nitrogen delivery is not a linear function of Nitrogen content in the leaf at least for the first look at the Nitrogen delivery. In the following sections to follow we will look how One Way ANOVA statistics can support this.
3 Its all about variation: Comparing several means by ANOVA
We can recall that the proponents of this study have decided for 7 treatments (incremental delivery increase of Nitrogen from 0-150% ) consisting of 4 replications (or four trials each on each treatment) which resulted to a response of 28 measurements from sample leafs all done by chemistry lab instrumentation. If there were fewer responses so that only two sets of data will be compared then, the comparison for variation will take on the statistics using the following:
• Comparison of two means: t-test for difference between two means
• Comparison of variance: F-test for test of precision between two data sets
Given the two data sets the above can be easily facilitated in Minitab c software using the Basic Statistics 2-sample t calculator for a two sample t-test. However, if there are several means to compare the proper statistic to use is what is called ”Analysis of Variance” or ANOVA. There is of course a test of the results from all of these from that they are indeed significant and that is the ”Test of Significance. A statistic based entirely by beginning a hypothesis (an educated and scientific guess) and at the end either accepting it or rejecting it based on statistical test.
3.1 What are the interesting questions: Null Hypothesis to ask?
We see in the previous section that well within a 5-10% decrease and 2.5-5% increase in Nitrogen from replication to replication is a possibility for all treatments. A small percentage will have impact if we are looking at bulk quantities for Nitrogen supply. Having a first look at the Nitrogen response the null hypothesis asks:
Ultimately the one factor experiment should lead to an answer in numbers for the following questions:
• What is the optimal level of Nitrogen delivery treatment if you only want to consider NDVI as a response from the UAV sensor?
• What is the optimal level of Nitrogen delivery treatment if you only want to consider one response only from leaf nutrient?
• What is the optimal level of Nitrogen delivery treatment if you want to consider only those main andinteracting effects from UAV sensor bands, VIs and Leaf nutrients lab results?
3.2 One Way ANOVA Results: Nitrogen vs. Treatment
We are now in the position to produce the results for this study which fits perfectly the scheme for design of experiment in which we have only one factor (Nitrogen Delivery) with multilevels of seven levels of delivery treatments. We arranged the data set in Minitab c such that there will only be two columns. The first column is the Factor (treatment) and the second column are the Observations (response). The set-up is shown as a screenshot in Figure 7 below:
We show above how to set-up Minitab c software for a One Way ANOVA for the data set that have been prearranged to allow the input requirement for the software. In column C1 the treatments A,B,C,D,E,F and G are arrange so that the replicates are the opposite column at C2. In the required input for a One Way ANOVA experiment it must be done in the manner shown in Figure 8.
Below we show the successive output of analysis and provide some explanation from results. Our first premise is that all of the means are equal for all treatments – this is our Null Hypothesis. If after our ANOVA run we found out otherwise then we must reject this hypothesis and accept for the alternative hypothesis. As the first line of our results show in Figure 9 below as expected the ANOVA run determined the factor is the Treatment and there seven levels coded as A,B,C,D,E,F and G as shown in Figure 10. In Figure 11 the ANOVA
Figure 9: Method is to compare the Means and accept or reject the Null Hypothesis
run calculated a P-value = 0.0005 which less than α=0.05 (default) this a significance level denoted as α that indicates a 5% risk of concluding that a difference exists when there is no actual difference. As shown in the Analysis of Variance table in Figure 11 provides enough evidence to conclude that the means of responses from the results of the treatment or nitrogen delivery are significantly different. Although already obvious from the basic regression line shown in Section 3.1 but this a confirmation a statistical proof or evidence.
Figure 10: Minitab reports correctly for one factor and seven levels
Figure 11: Minitab reports the important model summary
In Figure 11 above we take note the DF (degrees of freedom) from 6 to 21, the F-value is in between them with a very small value of 0.0005 (the F-distribution curve is not shown here). The model summary in Figure 12 run shows the R-sq is good to 64.98% and the standard deviation (pooled SD) is quite small.. We now take a look
Figure 12: Minitab reports the important model summary
at the results of the means calculation in our ANOVA run as shown Figure 13. We see that for each treatment the means have been calculated as well as their individual standard deviation (a pooled SD is no the same as the usual SD) at confidence interval of 95% according to NIST (US) provides a range of values which is likely to contain the population parameter of interest. A quick check for example for Treatment A with am mean value of 0.02696 is in the interval (0.17221,0.24529) which is likely to be found in this interval. It means also that it has a probability of 0.95 to be found in that interval. The ANOVA run can generate automatically an interval plot shown in Figure 14 for the Nitrogen per each treatment, the mean is marked with a dot between the interval and joined by a line. Here we see that the dots does not line up except at D and E but are otherwise increasing from treatment to treatment – hence, another indication that the difference in the means for this process are really different there we reject the Null Hypothesis H0 and accept H1.
Figure 13: Minitab reports the important model summary
Figure 14: Generated interval plot showing the location of the means per treatment
3.3 Process Model: Line fit Regression by One Way ANOVA
A process model can be described as polynomial with constant coefficients with independent variable (factors) on the right side of the equation and dependent variable (response) on the left side of the equation as shown in Figure 15. As we can see that regression is a line fit traversing the average of the data points. We take on a special case when truncate the treatment at the 5th stage or Treatment E when we take note that the rise from A to D seemed to plateau at at E shown in Figure 16 below: We can recall from Section 2 that a seconds order polynomial regression directly from the Nitrogen mean values also when it was truncated gives R2=0.97 which interest us to explore a possible optimal value at the point.
Figure 15: A process model for the Nitrogen delivery given a treatment
Figure 16: A process model for the Nitrogen delivery is truncated to Treatment E
We know that Nitrogen is the response from its delivery from different treatments as we perform the One Way ANOVA, we can rewrite the resulting fitted line plot as follows:
Y = 0.2199 + 0.001042 · X − 2 × 10−6 · X2 (1)
On the other hand for the special case of a truncated treatment in Figure 16 performing the same procedure we arrived at the following regression line:
Y = 0.2104 + 0.0002038 · X − 0.000013 · X2 (2)
For convenience we show the earlier derived two regression lines from Section 3.1 to compare with those derived by One Way ANOVA here. When the regression is linear:
Y = 0.2268 + 0.0008 · X (3)
When the regression is a 2nd order polynomial line fit:
Y = 0.2227 + 0.001 · X − 1 × 10−6 · X2 (4)
When the regression is a 2nd order polynomial line fit but intentionally truncated at 5th treatment in Figure 6 in Section 2 we have the following:
Y = 0.2132 + 0.002 · X − 1 × 10−5 · X2 (5)
Y – Nitrogen output in A.U.
X – Nitrogen input delivery treatment as per % Plantation Practice
Equations (1),(2), (3), (4) can be used to estimate the Nitrogen response beyond the last treatment scheme (here at G) or anything between A to G treatments. Equation (5) is a special case when it is decided that the last Treatment will be made at E (or 100% PP) will be accepted for the highest correlation.
4 Selecting the best model: Criteria of Cost Function
Numerically we could compare how far is the prediction model i.e.regression line with the actual point or points away from it in statistical terms we know that these are called residues as computed by ANOVA however, we can introduce another formulation to compare between the predicted and measurement lab results and this is looking at the what is called Cost Function which is defined by Equation 6 below:
C(y)min = Σ(yo − yp) 2 (6)
C(y)min – the cost function being the most minimum among the models being considered
yo – the actual output or response of the variable
yp – the predicted results from the model
>And so the process is quite simple, simulate the regression model to obtain the model numerical results and get the sum of the squared difference between the predicted versus the measured or actual response values. Comparing among the regression lines being considered the minimum cost will be the best model.
5 Optimal delivery of Nitrogen at the apparent plateau response
When one takes into consideration at some point of delivery of nitrogen particularly at 75% and when truncated at 100% we see that the response has seized to become monotonic as in the case of fitted line plots shown in Section 2 and in Section 3.3. If we take this case for a point of optimization then it is in the determining of the local maximum at vicinity where the critical point lies that is when Equation 7 rewritten below is when the first derivative is taken:
Y = 0.2104 + 0.0002038 · X − 0.000013 · X2 (7)
dX/dY = 0.0002038 + 2(0.000013) · X (8)
dX/dY = 0 (9)
we have for the critical point at:
X = −0.7838462 ≈ 78% (10)
Hence we have one global maximum at about 78% since we have a smooth function that entirely takes the graph of a concave.
6 Consideration of other nutrients and Vegetation Indices
There are 7 elements (Nitrogen is done here) from leaf nutrients and 19 vegetation indices from the UAV sensor response. The reader is urge to try out performing the One Way ANOVA for each and every elements there is as well as a select vegetation index for purposes of exercise and reinforcement of understanding the computational process. A screen shot of a possible output when performing One Way ANOVA from the rest of the responses.
7 Conclusions and Recommendations
We have shown that in the case of single factor that of Nitrogen Delivery and in seven levels of percentage application in four replicates the model that leads to a 2nd degree polynomial regression line. We also note that if the Nitrogen delivery is truncated at 100% we have the highest correlation possible which leads to a cubic regression line. When this is used we can infer from the fitted line plot that the optimum maybe in the region between 75% – 80% .
We would like to thank firstly the Agricultural Research Department of the Del Monte Philippines – Mindanao for their kind permission to report on this conference our computational method for this case study. With utmost gratitude to the guidance and review of Mr. Bernardino Buenaobra from the USC Phil-LiDAR Research Center from the University of San Carlos, Cebu City in the checking and carrying out of the statistical analysis from our field data.
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