1 Introduction

Elephant foot yam (Amorphophallus paeoniifolius) is a tropical tuber crop that has production potential. This is cultivated in different regions as a cash crop. Genetic background of yam across regions is studied e.g., in Santosa et al. (2017). Sensitivity of Elephant-foot-yam plants under stress is analyzed in relation to yield to have higher yield than usual. Plant stress may suitably be used to achieve higher yield of yam, e.g., see Dasgupta (2017a), Dasgupta (2017b). Yam plants are found to be stress resistant when cultivated in a harsh agro-climatic environment. Seed weight 800 g is seen to be appropriate for high yield in a study reported in Dasgupta (2017a). When the stress is extreme for plant survival, higher seed weight supports the plant growth at initial stage and 800 g of seed weight produced more yam in an experiment conducted in Giridih, Jharkhand (India).

Harsh agro-climatic environment in the field experiment with a few irrigations given in peak of summer in the beginning of the experiment acted as severe stress, the yam plants were further subjected to interim yam detachment at either of the two time zones in growth experiment and remaining stem structure was replanted as described in Dasgupta (2017a). The first interim yam detachment time was at two and half months after sprouting, and the second interim detachment time was at four and half months from sprouting. Yams are detached only once during the experiment from alive plants.

For seed weights 500 g, 650 g and 800 g; the second time period for interim yam detachment from plants is seen to be superior to achieve high yield in total.

Dasgupta (2017b) concentrated on second interim yam detachment strategy made at four and half months from sprouting, and analysed the growth curve in terms of almost sure confidence band, proliferation rate, and estimation of growth curve via mid band to contain fluctuations of curves around central line. Modeling the error component by a Gaussian process was investigated. Estimation of process parameters from observed data by different techniques including the method of maximum likelihood and from observed maximum fluctuation of the growth curves were investigated. Test for the hypothesis that the error components are following a Brownian motion is seen to be affirmative.

In the present study we analyse the data on three dimensional surface plotting, taking into account the variation of the time dependent proliferation rate with seed weight in a continuous scale. The peak of the proliferation surface starts with 500 g of seed weight at the front edge in a three dimensional figure. The said peak is towards the end of time range for yam lifetime, and then the line of peaks of curved surface goes downward through the interior of time range for other seed weights of higher values; the peaks of proliferation rates are attained inside the time range of lifetime for other seed weights more than 500 g. This suggest, if plants with seed weight 500 g could be nourished further after some time from sprouting, then yield from these would be much higher. Proliferation rates when integrated provides logarithm of yield. Higher the rates, higher are the yield in those time regions.

If additional resources are made available at a later stage, farmers may like to use that for nourishing the plants with seed weight 500 g in order to maximise yield in a harsh and extreme agro-climatic environment.

2 Materials and Methods

We studied longitudinal growth of Elephant-foot-yam for sixty plants under extreme agro-climatic stress in a field experiment conducted in the agricultural farm at Indian Statistical Institute, Giridih, with seed weights 500 g, 650 g and 800 g of yam.

The experimental layout consists of six columns, in each column there are ten equidistant pits at a distance of 1 m. First two columns are for seed weight 500 g, next two are for seed weight 650 g, and the last two columns are for plants with seed weight 800 g. Column to column distance is also 1 m; the plants are numbered 1-10 in the first column, 11-20 in the second etc, see Dasgupta (2017a).

We analyse the characteristics derived from longitudinal growth curves in three dimensional analyses. The earlier studies reported in Dasgupta (2017a), Dasgupta (2017b) indicated that yam detachment at the time of second interim growth recording is superior. The longitudinal growth curves of yam growth from 31 plants over different seed weights, all subjected to interim yam detachment at the time of second growth recording are shown in Figure 1. The figure appears as the first figure in Dasgupta (2017a). The Mean growth curves for different seed weights are computed as mean of y coordinates in Figure 1, for fixed values of time x; when at least one growth data is recorded in the field experiment at that time. The mean of y values are connected by straight lines as an approximation of mean response and shown in Figure 2. Starting with this, we compute the proliferation rates i.e., the derivative of logarithm of yam yield with respect to time, in each group of plants with fixed seed weight viz., 500 g, 650 g, 800 g. The computational procedure involves raw proliferation rates computed at fixed time t. A smooth exponentially decaying weight function, that down-weights the raw rates, involving distant observations, away at long time from t; is considered next. Averaging these weighted rates, or taking suitably trimmed mean of these weighted rates are two possible options for consolidation in a single number. The latter option considered previously produced too smooth rates. The former option of averaging and then smoothing the averaged rates with the program smooth.spline in SPlus is considered as a modification; the procedure is described in Dasgupta (2017c), and seen here to produce satisfactory result for proliferation rate over time, including extreme time points. On the other hand, median or suitably trimmed mean of these weighted rates considered earlier, although perform well otherwise, performs poor at extreme time points. We then concentrate on the time region where the proliferation rate is high for a seed weight; this indicates rapid growth of underground yam in that time segment, enabling comparison.

The longitudinal growth curves of yam yield with interim detachment of yam at the time of second growth recording are shown for different seed weights in Figure 1. This is same as Figure 1 of Dasgupta (2017a). There are 8 plants of 500 g seed weight, 13 plants of 650 g seed weight, and 10 plants of 800 g seed weight subjected to interim yam cut at the time of second growth recording. Second interim cut of yam along with final harvest on maturity from replantation of stem structure, with a few attached roots in the same pit produced higher yield in total. Further analysis of the data has to be made for plants with second interim cut. In general, curves with seed weight 800 g are seen above other curves. 800 g seed weight corresponds to higher yield in harsh and extreme agro-climatic environment. For seed weight 500 g, sharp upturn of all longitudinal growth curves is observed beyond a time from sprouting till the end, indicating possibility of these curves crossing the other growth curves from below over time, if lifetime of plants with seed weight 500 g could be extended further.

Consider a fixed seed weight and the relevant plants with yam cut at the time of second interim growth recording. For each plant alive, growth recordings are available at two interim time points of uprooting, along with initial and final weight at mature stage harvesting. y co-ordinate of curves in Figure 1, with different time points x for which yield data from first interim, second interim, final harvest are available in records are considered. For such fixed time x from the above set, mean of y values lying in the curves, obtained from linear interpolation, provides an estimate of mean response at that time point. The mean values joined by lines provide a raw estimate of mean response curve for a particular seed weight in Figure 2. Mean curves for different seed weights are of use to compute proliferation rates. Seed weight 800 g corresponds to the highest yield. Mean curve of yield for seed weight 650 g and 800 g seem a bit erratic towards end due to scarcity of observations therein at far end. Mean curve corresponding to seed weight 500 g has upward slope from 87 days onward, approximately after 3 months; the growth rate goes on increasing to a steeper level especially after 125 days and continues till the end of lifetime. If the plants with seed weight 500 g are properly nourished after a time gap from sprouting, then there is a possibility of higher yield from these plants.

A three dimensional representation of proliferation rates drawn in R with the program `persp', taking independent variables as seed weight and time, makes the comparison more transparent.

Figure 3, Figure 4, Figure 5, Figure 6,and Figure 7 show longitudinal growth characteristics of 31 plants at different time points when growth data were collected; the growth values are linearly interpolated in the case growth data is not available for a plant at a time point (measured from sprouting time of the plant), when another plant in the experiment had growth data recorded at that time (measured from sprouting time of that plant). Sprouting times are in general different for different plants. Three dimensional figures, except the one based on seed weight, are computed on interpolated values given in Figure 3, Figure 4, Figure 5, Figure 6, and Figure 7. Two dimensional figures are based on raw data collected on plants; not on interpolated values given in Figure 3, Figure 4, Figure 5, Figure 6, and Figure 7.

Least squared regression analysis is done to identify the significant variables in the set of seed weight (x₁), time (x₂), maximum plant height (x₃), maximum girth at base (x₄), maximum girth at middle (x₅), maximum girth at top (x₆), and canopy radius (x₇).

If seed weight is not known, or not recorded then the variables plant height, canopy radius, or stem girth may be considered as replacement of seed weight in the three dimensional representation. These pictures provide a deeper insight into the yam growth process, even when seed weight is recorded, thus enabling decision making on the basis of above ground auxiliary variables, which are highly associated with yam yield.

3 Results

3.1 Regression analysis of yield on associated variables when yam detachment is made at second yield recording

Analysis of variance in linear model on longitudinal data from 31 plants with interpolated values of variables at interim time points are shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 with yield y=y(t), t ɛ [0, 203] days and seven auxiliary variables associated with interim weight y of yam. The variables in the Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 are seed weight (x₁), time (x₂), maximum plant height (x₃), maximum girth at base (x₄), maximum girth at middle (x₅), maximum girth at top (x₆), and canopy radius (x₇) and yield y(t) at time t. The values of the variables are linearly interpolated at intermediate time points.

At time t=0, proliferation rate is taken to be 0. The data so constructed are used in R program with package `persp' for three dimensional plots.

Data given in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 provide the following least squared regression line with R²=0.4638.

y = - 0.5955137 + 1.3842935 x₁+ 0.0025382 x₂ + 0.0044606 x₃- 0.0086307 x₄ + 0.0278280 x₅-0.0505629 x₆+ 0.0030111x₇

Of these, the variables seed weight x₁ and time x₂ are highly significant with respective p values 7 e-15 and 1.07 e-09, in t test.

3.2 Proliferation rates of yield versus seed weight

Proliferation rate is related to yield by the relation y (t) = exp (), higher the value of rate over a considerable time region, higher is the yield. We compute the proliferation rate curve by a modified technique proposed in Dasgupta (2017c), in Figure 8, Figure 9,and Figure 10, for seed weight 800 g, 650 g, and 500 g, respectively. Figure 8 has a hump like structure indicating high rate over a considerable region of time, indicating massive accumulation of yam underground in that time zone. The performance is slightly poor for seed weight 650 g in Figure 9, as the peak value is not maintained for long. Proliferation rate of yield for seed weight 500 g rises steep onto the top of Figure 10, indicating a good potential of yam growth if the rate could be maintained in an extended lifetime by nourishing the plants.

Proliferation rate    is a scaled version of velocity   . This measure is independent of the choice of unit used in measuring y. Consider (x, y) coordinates of mean growth curve in Figure 2 for 800 g of seed weight with yam detachment at the time of second growth recording. Time points x when at least one data point from first interim, second interim, final harvest are available from plants with seed weight 800 g. Lowess regression is a nonparametric smoothing technique, see e.g., Cleveland (1981). These set of (x, y) points is now lowess regressed at the first stage with f = 0.35 to obtain smoothed (x, y) values in computation of proliferation rates. For yam yield y with seed weight 800 g, and growth curve shown in Figure 2, the proliferation rate is obtained in Figure 8. Computation is based on a technique proposed in Dasgupta (2017c), with exponentially decaying weights attached to empirical slopes computed from data pairs at different time points with respect to a fixed time point t of interest. More weights are given to data points near the time t in derivative computation, and less weights are assigned to distant time points from t. Weighted mean of these empirical slopes at derivative stage is divided by y, and the empirical rates are smoothed by smooth.spline with spar = 0.00001 in SPlus to obtain proliferation rate at time point t. The curve in Figure 8 has a sharp upturn from the beginning, then it remains relatively steady with high values of proliferation rate in a region near the peak that looks like a hump; and then the curve gradually decreases towards zero with progress of time.

The proliferation rates for seed weight 650 g is drawn in a similar manner like Figure 8. The curve in Figure 9 reaches a peak at a later time point compared to that for seed weight 800 g. The peak is sharp unlike that for 800 g. The rise is slow to the peak and the fall from peak towards zero is sharp.

The proliferation rates for seed weight 500 g shows increasing trend over almost the entire period of lifetime. The rate is sharper after 100 days, and becomes steady towards the end. Starting with negative proliferation rate at start, these plants seem to gradually adapt with harsh agro-climatic environment with progress in time. If the same rate could be maintained in an extended plant lifetime with supporting nourishments to these plants, then yam yield from these could be much higher, although 500 g is the least seed weight considered in the experiment.

3.3 Three dimensional figures on proliferation rates

Three dimensional figures provide a deeper insight into the yam growth process compared to 2-dimensional plots, on variation of rates with simultaneous variation in two auxiliary variables associated with yield on continuous scale.

In Figure 11 proliferation rates of yield vs. time and seed weight are shown. This picture is based on raw data collected, enabling comparison with two dimensional Figure 8, Figure 9, and Figure 10. Seed weight 500 g has highest value on the surface of proliferation rate. The other peak of the curve at far end, corresponding to the seed weight 800 g is in a slanted surface downward, these seed weights correspond to lower values of peaks in proliferation surface. For all other seed weights the peaks of the rates are attained inside the time range. Seed weight 500 g has highest value of proliferation rate in Figure 10. The same is expected to be seen on the surface of proliferation rate in Figure 11.

In Figure 12 proliferation rates of yield vs. time and plant height are shown. The figure reveals that high value of proliferation occurs at about 120 days for plants of average height. If the plant height is high in less time, then the rate is high. The same is true, if plant height is moderate at large value of time.

In Figure 13 proliferation rates of yield vs. time and canopy radius are shown. High values of proliferation occur for moderate value of canopy radius. For canopy radius around 40 cm, rate is usually high after a time lapse from sprouting. For large values of canopy radius, proliferation rate attains high values relatively early.

For plants with fixed seed weight undergoing second interim cut, we computed the coefficient of variation (c.v.) for the stem perimeters: maximum girth at base, maximum girth at middle, maximum girth at top of stems; maximum is also taken over lifetime of plants. It turned out that c.v. of `maximum girth at top' are the highest amongst recordings of girths on base, middle and top of stem, within each group of seed weight 500 g, 650 g, 800 g; the highest values of c.v. in the respective groups are 16.79542104, 12.58080459, 15.77286043. All these correspond to the girth at top of stems in plants, indicating maximum variation of girth at stem top, in collected data for analysis. Consequently, out of measures on girth at three locations, we shall consider girth at the top of stem to draw 3 - dimensional picture to study variation of proliferation rate with respect to girth.

In Figure 14 proliferation rates of yield vs. time & maximum girth at top of stem are shown. Proliferation rate is high in general, when girth is around 6 cm. In the middle of lifetime, if the girth is approximately 5 cm then the rate is high. The upper edge of proliferation surface corresponds to around 120 days.

When seed weights are not recorded, Figure 12, Figure 13, and Figure 14 may shed light on the plant growth status on yield.

We plot a three dimensional figure on rate vs. time and seed weight in Figure 11. Recall the construction of Figure 2, consider a fixed seed weight and consider the relevant plants with yam cut at the time of second interim growth recording. For plants with a fixed seed weight, y co-ordinate of curves over different time points x, whenever at least one data point from first interim, second interim, final harvest are available in a curve, that time point was considered in Figure 2. Now for a fixed seed weight, compile all these time points, counted from the time of sprouting in relevant plant, where at least one yield data is recorded. Different points on time and rate   (t, r (t)) computed via the program in Dasgupta (2017c) are obtained for plants with fixed seed weight, and Figure 8, Figure 9, Figure 10 are based on these information.

For a particular seed weight, proliferation rate at an interim time point in the above set of time t is linearly interpolated from the points (t, r (t)) of that seed weight having immediate upper and lower value of time. This is possible because each group of plants with a fixed seed weight has growth records at time zero at sprouting, 1st interim growth recording time, second interim growth recording time, and time at final harvest. Lowess smoothing of these values was the first step in computing proliferation rates in Figure 8, Figure 9, and Figure10.

With these data on rates, at times interpolated, specific to each seed weight; a three dimensional picture of proliferation rate of yam yield with time and seed weight is drawn in R by the program `persp' and shown in Figure 11. Seed weight 500 g has highest value on the surface of proliferation rate in Figure 11. Over time, the curved surface rises from low values near zero and reaches to the line of peaks in a valley like structure, within an interior region of time range before the smooth surface gradually lowers itself down; except for the single curve seen at fore front which is nearly steady around peak; the front edge curve corresponds to seed weight 500 g. The other peak of the curve at far end, corresponding to the seed weight 800 g is in a slightly slanted surface downward, have lower values of peaks in proliferation surface, correspond to all other seed weights in continuous scale, and these peaks of the rates are attained inside the time range considered.

We also checked drawing of proliferation surface like that of Figure 11 based on interpolated proliferation rate for each plant. Linear interpolation of proliferation for each plant in a group of fixed seed weight, with a larger set of time points where at least one growth record is available covering plants in all seed weights produced not so satisfactory result, violating some features of Figure 8, Figure 9, and Figure 10 based on original recorded data. This is possibly due to estimation of plant specific proliferation rate at too many interim points with limited existing data.

In a similar manner of drawing Figure 11, we draw Figure 12. Recall the set of time points considered in constructing Figure 11 where at least one yield data is recorded. Over these time points, we consider plant height for a fixed seed weight. There are height records at first interim growth recording time, second interim time, and the last height on plant maturity. In total there are at least 5 observations on height recorded for each plant. The set of time points considered for plant height recording is larger than that of underground yield, as height is an aboveground variable easy to measure. Proliferation rate in these interim time points is linearly interpolated from time points of recorded growth data on that plant of specified seed weight and associated proliferation rates. The proliferation rates are available for fixed seed weight and these are used in drawing Figure 8, Figure 9, and Figure10.

If a plant height is recorded at a relevant time point, then that is taken into account. Otherwise the plant height at that time point is interpolated from immediate upper and lower value of time when height is recorded in that plant.

For the same time point counted from sprouting, there may be several different heights from different plants with a fixed seed weight; we have taken all these different heights into account in the R program. A three dimensional picture of proliferation rate of yam yield with time and plant height for plants with yam detachment at second interim growth recording reveals that high value of proliferation occur for general plant height at about 120 days. If the plant height is high in less time, the rate is high. The same is true if plant height is moderate in large value of time. Plant height usually goes up with time and then it decreases slowly with time when plants mature at old age. When seed weight of the yam corms planted are unknown or not recorded, then the above picture is of help to ascertain the growth status of yam underground, based on the observable plant height.

Canopy radius of yam plants is an easy to measure and important variable that is sensitive to plant stress. In a similar manner of drawing Figure 11, Figure 12, we draw Figure 13 on proliferation rate of yield vs. time & canopy radius. At a particular time point there may be several different canopy radii for different plants with fixed seed weight, we have taken all those different canopy radii into account in the R program. A three dimensional picture of proliferation rate of yam yield with time and canopy radius for plants with yam detachment at second interim growth recording reveals that high value of proliferation occur for moderate value of canopy radius. For canopy radius around 40 cm, rate is moderately high after a time lapse from sprouting. For large canopy radius, rate attains high values relatively early. Canopy radius usually goes up fast with time in the beginning and then it decreases slowly over time when plants mature at old age. Like the earlier stated situation, when seed weight of the yam corms planted are unknown or not recorded, the Figure 13 is of help to ascertain the growth status of yam underground, based on observable aboveground variable canopy radius.

Maximum girth at the top of stem, over different stems in a plant is also easy to measure and coefficient of variation of this variable is highest, when measured at three locations viz., top, middle, and base of stems. In a similar manner of drawing Figure 11, Figure 12, Figure 13, we draw Figure 14. Proliferation rate is high in general, when girth is around 6 cm. In the middle of lifetime, if the girth is approximately 5 cm, then the rate is high. The upper edge of proliferation surface showing line of peaks corresponds to time 110 days. Figure 14 is of similar pattern with that of Figure 13. Top of the stem is a well defined position to measure girth. In contrast, base position is affected by variation over time in level of ground soil, which also affects the middle position in a stem while taking growth readings.

4 Discussions

Plant stress may suitably be utilized to increase yield in some cases. Yam plants can adapt to extreme stress when the seed weight is high. Earlier studies conducted in Giridih, Jharkhand (India) indicated that interim yam detachment at about four and half months from sprouting and subsequent replantation of stems with a few roots attached at the base of stems from seed weight 800 g has an increasing effect on total yield in extremely harsh agro-climatic environment. Plants with seed weight 500 g have a steep proliferation rate of yam yield after a time gap from sprouting. This suggests a possibility of using small seed corms for high yield even when the plant survival condition is not so conducive. This may be accomplished by providing extra nourishments to these plants with seed weight 500 g, near the maturity period.

5 Conclusions

Plants with seed weight 500 g is seen to have steep proliferation rate after a time gap from sprouting among seed weight in the range 500 g - 800 g, when the plants are uprooted at about four and half months from sprouting time and underground yam is detached before replanting the stems with a few roots attached to stem and experiment continued till plant maturity. The procedure produced high yield of yam in total. A possibility of higher yield in extremely harsh agro-climatic environment may be explored by providing extra nourishments after a time gap from sprouting to the plants with seed weight 500 g having high proliferation rates of yam yield.

Refrences

Cleveland W.S., 1981, LOWESS: A program for smoothing scatter plots by robust locally weighted regression, The American Statistician, 35 (1): 54

Dasgupta R., 2015, Rates of convergence in CLT for two sample u-statistics in non iid case and multiphasic growth curve, growth curve and structural equation modeling, In: Dasgupta R (ed.) Proceedings in mathematics and statistics, vol 132, Springer, Berlin, pp 35-58

Dasgupta R., 2017a, Longitudinal Growth Curve of Elephant Foot Yam under Extreme Stress and Plant Sensitivity. International Journal of Horticulture, 2017, Vol. 7, No. 13

Dasgupta R., 2017b, Longitudinal Growth Curve of Elephant Foot Yam under Extreme Stress and Plant Sensitivity II, Growth Curve Models and Applications Springer (USA). Chapter10. (To appear)

Dasgupta R., 2017c, Longitudinal studies on Mathematical Aptitude and Intelligence Quotient of North Eastern tribes in Tripura, Advances in Growth Curve and Structural Equation Modeling: Proceedings 2017 Springer (USA), Chapter 1. (To appear)

Santosa E., Lian C.L., Sugiyama N., Misra R.S., Boonkorkaew P., Thanomchit K., 2017, Population structure of elephant foot yams (Amorphophallus paeoniifolius (Dennst.) Nicolson) in Asia, PLoS ONE 12(6): e0180000

https://doi.org/10.1371/journal.pone.0180000