# Hidden Markov Models for Real-Time Estimation of Corn Progress Stages Using MODIS and Meteorological Data

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data Sets

- (1)
- Daily NDVI time series, which is derived from the atmospherically corrected MODIS MOD09GQ (MODIS Surface Reflectance Daily L2G Global 250 m) dataset with 250 m spatial resolution. This data set is publicly available through the “Vegetation Condition Explorer” ( http://dss.csiss.gmu.edu/NDVIDownload/), maintained by the Center for Spatial Information Science and Systems (CSISS), George Mason University.
- (2)
- NASS’s Cropland Data Layer (CDL), which is a raster, geo-referenced crop-specific land use data layer. The spatial resolution of years 2006–2009 is 56 m, and the rest is 30 m. The data set is publicly available via “CropScape” ( http://nassgeodata.gmu.edu/), produced operationally by USDA/NASS.
- (3)
- NASS’s CPRs, which record the percent complete (area ratio) of crop fields that has either reached or completed a specific progress stage over a specific administrative unit. It is publicly available via NASS’s “Quick Stats 2.0” service ( http://www.nass.usda.gov/Quick_Stats/). More details of the first three data sets can refer to [19].
- (4)
- Daily minimum and maximum temperatures, which are derived from the United States Historical Climatology Network (USHCN) [27]. USHCN is a high-quality network of US Cooperative Observer Network stations, specially selected for analyzing long-term variability and change in the whole contiguous United States [27]. In this study, 23, 33, and 37 meteorological stations are chosen for the states of Iowa, Illinois, and Nebraska, respectively (Figure 1 and Appendix: Table A1). The meteorological stations were selected with a number of criteria including length of period of record, and spatial coverage.

## 3. Feature Extraction

#### 3.1. Mean NDVI

#### 3.2. Fractal Dimension

#### 3.3. AGDDs

_{min}(t) and T̄

_{max}(t), respectively. In addition, to calculate the AGDDs, adjustment should be performed on T̄

_{min}(t) and T̄

_{max}(t) by the rules of corn response to temperature stress. The adjusted T̄

_{min}(t) and are defined as T

_{min}(t) and T

_{max}(t), respectively.

_{min}(t) and T̄

_{max}(t) are generated by weighted average of all available meteorological stations within the corresponding administrative border. The Thiessen polygon approach [30], a geospatial technique, is applied to graphically weight meteorological station data. In this approach, each station is weighted in direct proportion based on its area of influence in the total area of specified administrative unit. It assumes that any point of temperature condition is equal to that of the nearest station. We take the calculation of T̄

_{min}(t) as example, and T̄

_{max}(t) is the same. The T̄

_{min}(t) over a state is calculated by

_{i}is the weight of station i, which can be determined by its corresponding influence area of station i, i.e., w

_{i}=A

_{i}/A

_{total}, where A

_{total}is the administrative area of the given state, and A

_{i}represents the influence area of station i that is divided by Thiessen polygon. The methods we used for calculating T̄

_{min}(t) in this paper are especially applicable and useful to avoid the ambient temperature data, which may be missing from the time series of records in actual practice.

_{base}, and ceases as temperatures exceed an upper threshold [31]. In the United States, the usual low temperature stress of corn or base temperature T

_{base}is 10° [32]. In addition, previous studies have shown that corn growth slows at temperatures above 30° [32]. Therefore, we use 10° and 30° to adjust T̄

_{min}(t) and T̄

_{max}(t) accordingly. That is, if the lowest temperature for a day is below the 10°, then 10° is used as the T

_{min}(t), and if the highest temperature is over the 30°, then 30° is used as T

_{max}(t). Start date is set as 1 April. Given the T̄

_{min}(t) and T̄

_{max}(t), and according to the rule of adjustment, the AGDDs at the DOY can be calculated by

## 4. Corn Progress Percentages Estimation

#### 4.1. Specifying an HMM

_{1}, …, S

_{N}}, and T observation sequence O = {O

_{1}, …, O

_{T}}. In this study, the hidden stages consist of pre-season (S

_{1}), planted (S

_{2}), emerged (S

_{3}), silking (S

_{4}), dough (S

_{5}), dent (S

_{6}), mature (S

_{7}), and harvested (S

_{8}). The pre-season stage, which represents the period when corn hasn’t been planted, is added as the first time interval to facilitate the design of the model. Let q

_{t}, (t = 1, …, T) be a variable of the hidden stage at time t. For example, progress stage S

_{i}, (i = 1, …, N) at time t is denoted by q

_{t}= S

_{i}. Therefore, we can specify an HMM of a corn progress by its parameters λ = (A, B, Π), where A is the stage transition probability matrix whose entry, a

_{i,j}(t) = P(q

_{t}= S

_{j}|q

_{t}

_{–1}= S

_{i}), (i, j = 1, …, N) determines the transition probability from stage S

_{i}to stage S

_{j}at time t; B is the observation probability matrix whose entry, b

_{j}(O

_{t}) = P(O

_{t}|q

_{t}= S

_{j}), indicates the probability that the observation O

_{t}are generated by the stage S

_{j}at time t; Π is the initial probability distribution whose entry, π

_{i}= P(q

_{1}= S

_{i}), determines the probability of the model being initially in stage S

_{i}at the first time node (i.e., t = 1). π

_{i}also represents the prior probability of stage S

_{i}at time t = 1. It can be extended by π

_{i}(t) = P(q

_{t}= S

_{i}) that represents the prior probability of S

_{i}at time t. The joint probability distribution over all of the variables is given by

_{1,}…, q

_{t}is assumed to be a typical Markov chain with a first-order Markov assumption, i.e., stage at q

_{t}can only be decided by stage of previous latent variable q

_{t−1}and independent of all other stages. We abbreviate P(q

_{t}= S

_{j}|q

_{t−1}= S

_{i}) as a

_{i,j}(t). In addition, the observation O

_{t}at time t can only be determined by its corresponding stage S

_{j}. P(O

_{t}|q

_{t}= S

_{j}) is abbreviated as b

_{j}(O

_{t}). Thus, the probability that mentioned in Equation (3) is also equal to

#### 4.2. Mixture Model in HMMs

_{i}(O

_{t}), i.e., each stage of discrete variable q

_{t}represents a different component. The probability of observation is given by

_{i}(t) can be regarded as the weight of the ith component, and ${\sum}_{i=1}^{N}{\pi}_{i}(t)=1$.

_{i}(O

_{t}) is given by continuous probability density functions, i.e., Gaussian distribution.

#### 4.3. NASS’s CPRs Normalization

_{i}at time t noted as ${p}_{i}^{t}$, (i = 1, …, N). Ratios represent stages complete, rather than the proportion of each stage occupancy over an administrative unit in current time (Figure 4(a)). Corn phenological stages are unimodal in the life cycle. For a single corn plant, the arrival of S

_{j}, (1 ≤ i < j ≤ N) means S

_{i}has already completed. That is, S

_{i}is nested within S

_{j}, e.g., 19% of dough (S

_{5}) has completed means at least 19% of silking (S

_{4}) had completed already.

_{i}(t) can be straightforward to signify the area ratio of stage S

_{i}occupancy at time t for a specific administrative unit. Theoretically, π

_{i}(t) can be calculated from the original recording of NASS’s CPRs. It should be noted that some stages have no records of NASS’s CPRs data, because stage has not arrived or even passed by. Thus, before the π

_{i}(t) calculation, we need a data filling process for CPRs data. If the data recorded has not reached a certain stage, the value is set to 0, and if the developmental stage has passed, then the value is set to 1.

_{3}) takes at least 9.5 days (bigger than a week) delays after the planted (S

_{2}). Then, π

_{i}(t) is calculated by

_{6}), dough (S

_{5}) and silking (S

_{4}) have completed 1%, 19% and 96%, respectively. Based on Equation (6), we know that this region has 1%, 18%, 77% and 4% of corn plants at dent, dough silking and emerged (S

_{3}) stages, respectively (Figure 4(b)).

#### 4.4. HMM Parameters Determination

#### 4.4.1. Initial Probability Distribution

#### 4.4.2. Stage Transition Probability Matrix

_{i,j}(t) varies when time t changes. We assume a life cycle is unimodal, i.e., stage S

_{i}only can transform to itself or its next stage S

_{i}

_{+1}(Figure 5). a

_{i,j}(t) can be calculated directly from normalized NASS’s CPRs data. Thus, a

_{i,j}(t) is calculated by

_{t−1}= S

_{6}, then all transitions except a

_{6,6}(t) and a

_{6,7}(t) are zero element. a

_{6,6}(t) and a

_{6,7}(t) respectively correspond to the second and third restrictions, which sum up to 1.

#### 4.4.3. Observation Probability Matrix

_{i}(t). Probability density function associated with observations for each administrative unit can be modeled by a multivariate Gaussian distribution. Thus, in Equation (5), P(O

_{t}) is a linear superposition of Gaussian distribution, and b

_{i}(O

_{t}) is parameterized on mean vector μ

_{i}and covariance matrix ∑

_{i}.b

_{i}(O

_{t}) is given by

_{i}and ∑

_{i}are global HMM parameters, i.e., they are independent to time. The ith component weight (or mixing coefficient) π

_{i}(t) knowns from ground surveying, i.e., only μ

_{i}and ∑

_{i}are unknown. Given an observation sequence O

_{1}, …, O

_{T}, we can determine μ

_{i}and ∑

_{i}using maximum likelihood. The log-likelihood function with parameter space Θ = {μ, ∑}is given by

_{i}and ∑

_{i}and iteratively performs these two steps until convergence to a local maximum of the likelihood function. In the (q + 1)th iteration, the ${\mu}_{j}^{q+1}$ and ${\sum}_{j}^{q+1}$ are calculated by

_{i}and covariance matrix ∑

_{i}during the last iteration.

#### 4.5. Progress Percents Estimation

_{t}= S

_{j}|O

_{1}, …, O

_{t}). This is an online process (real-time), and can be solved by filtering based algorithms. We should emphasize that filtering, smoothing (offline), and prediction problems all compute the probability of hidden stages for given observations, e.g., P(q

_{t}= S

_{j}|O

_{1}, …, O

_{h}). More specifically, the difference is the smoothing problem compute by t < h, the filtering t = h, and prediction t > h. We note P(q

_{t}= S

_{j}, O

_{1}, …, O

_{t}) as κ

_{j}(t), which represents the probability of all the observation up to time t and the stage at time t is S

_{j}, then

_{j}(t) for the observation sequence of increasing interval t. Then, κ

_{j}(t) can be obtained recursively according to

_{j}and initial observation O

_{1}

_{t}= S

_{j}|O

_{1}, …, O

_{t}) which also represents area ratio of stage S

_{j}occupancy at time t for a specific administrative unit, an inverse normalize transfer process should be deployed. We note δ

_{i}(t) as the progress percent of stage S

_{i}at time t. Then it can be calculated by

## 5. Results and Discussions

#### 5.1. RMSE Results

#### 5.2. Accuracy Comparison

#### 5.3. Performance and Analysis

- (1)
- The accuracy of NASS’s CPRs. The NASS’s CPRs are surveyed data, and mainly depended on the subjective assessment of investigators. Thus, a bias error is inevitably introduced in the NASS’s CPRs data [41];
- (2)
- The quality of MODIS NDVI. Noise has inevitably disturbed the daily MODIS-NDVI images, e.g., cloud cover, missing data, mixed pixels, or some of the systematic errors that reduce the index value of daily MODIS-NDVI images;
- (3)
- The reliability of meteorological data, regarding to the observation data of weather stations, data missing, instrumentation, or observation station location change may affect the data homogeneity and spatial coverage;
- (4)
- Irregularities in raining and temperature pattern in different years, e.g., extensive drought occurs in a particular year, can significantly affect the stability of results. It would specially impacted on HMM parameters training, e.g., the stage transition probability matrix.
- (5)
- The insufficiency of temporal resolution. The temporal resolution of data is an important factor that affects the accuracy of corn progress stages estimation. As shown in Figure 4, the emerged stage just 9.5 days delays to planted stage, and dent stage approximate 15.4 days delays to dough stage. Accurate distinction between these growth stages requires a higher temporal resolution. It is really intractable that we have to trade off temporal resolution and data quality.

## 6. Conclusion

## Acknowledgments

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## Appendix

**Table A1.**Basic information of selected meteorological stations. Meteorological data include ID, name, and the geographic coordinate (i.e., latitude, longitude, and elevation) of each station. SA means US state abbreviations.

No | ID | SA | Name | Lat (°N) | Lon (°W) | Elev (m) | No | ID | SA | Name | Lat (°N) | Lon (°W) | Elev (m) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 130112 | IA | ALBIA 3 NNE | 41.07 | 92.79 | 268.2 | 48 | 116579 | IL | PANA 3E | 39.37 | 89.02 | 213.4 |

2 | 130133 | IA | ALGONA 3 W | 43.07 | 94.31 | 377.6 | 49 | 116610 | IL | PARIS WTR WKS | 39.64 | 87.69 | 207.3 |

3 | 130600 | IA | BELLE PLAINE | 41.88 | 92.28 | 246.9 | 50 | 116910 | IL | PONTIAC | 40.89 | 88.64 | 198.1 |

4 | 131402 | IA | CHARLES CITY | 43.08 | 92.67 | 309.1 | 51 | 117551 | IL | RUSHVILLE | 40.12 | 90.56 | 201.2 |

5 | 131533 | IA | CLARINDA | 40.72 | 95.02 | 298.7 | 52 | 118147 | IL | SPARTA 1 W | 38.12 | 89.72 | 163.1 |

6 | 131635 | IA | CLINTON #1 | 41.79 | 90.26 | 178.3 | 53 | 118740 | IL | URBANA | 40.08 | 88.24 | 219.8 |

7 | 132724 | IA | ESTHERVILLE 2 N | 43.43 | 94.82 | 396.8 | 54 | 118916 | IL | WALNUT | 41.55 | 89.6 | 210.3 |

8 | 132789 | IA | FAIRFIELD | 41.02 | 91.96 | 225.6 | 55 | 119241 | IL | WHITE HALL 1 E | 39.44 | 90.38 | 176.8 |

9 | 132864 | IA | FAYETTE | 42.85 | 91.82 | 344.4 | 56 | 119354 | IL | WINDSOR | 39.44 | 88.6 | 210.3 |

10 | 132977 | IA | FOREST CITY 2 NNE | 43.28 | 93.63 | 396.2 | 57 | 250130 | NE | ALLIANCE 1WNW | 42.11 | 102.9 | 1,217.4 |

11 | 132999 | IA | FORT DODGE 5NNW | 42.58 | 94.2 | 347.5 | 58 | 250375 | NE | ASHLAND NO 2 | 41.04 | 96.38 | 326.1 |

12 | 134063 | IA | INDIANOLA 2W | 41.37 | 93.65 | 287.1 | 59 | 250435 | NE | AUBURN 5 ESE | 40.37 | 95.75 | 283.5 |

13 | 134142 | IA | IOWA FALLS | 42.52 | 93.25 | 344.4 | 60 | 250640 | NE | BEAVER CITY | 40.13 | 99.83 | 658.4 |

14 | 134735 | IA | LE MARS | 42.78 | 96.15 | 364.2 | 61 | 251145 | NE | BRIDGEPORT | 41.67 | 103.1 | 1,117.4 |

15 | 134894 | IA | LOGAN | 41.64 | 95.79 | 301.8 | 62 | 251200 | NE | BROKEN BOW 2 W | 41.41 | 99.68 | 762 |

16 | 135769 | IA | MT AYR | 40.71 | 94.24 | 359.7 | 63 | 252020 | NE | CRETE | 40.62 | 96.95 | 437.4 |

17 | 135796 | IA | MT PLEASANT 1 SSW | 40.95 | 91.56 | 222.5 | 64 | 252100 | NE | CURTIS 3NNE | 40.67 | 100.49 | 829.4 |

18 | 135952 | IA | NEW HAMPTON | 43.05 | 92.31 | 349.9 | 65 | 252205 | NE | DAVID CITY | 41.25 | 97.13 | 490.7 |

19 | 137147 | IA | ROCK RAPIDS | 43.43 | 96.17 | 411.5 | 66 | 252820 | NE | FAIRBURY 5S | 40.07 | 97.17 | 411.5 |

20 | 137161 | IA | ROCKWELL CITY | 42.4 | 94.63 | 364.2 | 67 | 252840 | NE | FAIRMONT | 40.64 | 97.59 | 499.9 |

21 | 137979 | IA | STORM LAKE 2 E | 42.63 | 95.17 | 434.3 | 68 | 253175 | NE | GENEVA | 40.53 | 97.6 | 496.8 |

22 | 138296 | IA | TOLEDO 3N | 42.04 | 92.58 | 289.3 | 69 | 253185 | NE | GENOA 2 W | 41.45 | 97.76 | 484.6 |

23 | 138688 | IA | WASHINGTON | 41.28 | 91.71 | 210.3 | 70 | 253365 | NE | GOTHENBURG | 40.94 | 100.15 | 787.9 |

24 | 110072 | IL | ALEDO | 41.2 | 90.75 | 219.5 | 71 | 253615 | NE | HARRISON | 42.69 | 103.88 | 1,478.3 |

25 | 110187 | IL | ANNA 2 NNE | 37.48 | 89.23 | 195.1 | 72 | 253630 | NE | HARTINGTON | 42.62 | 97.26 | 417.6 |

26 | 110338 | IL | AURORA | 41.78 | 88.31 | 201.2 | 73 | 253660 | NE | HASTINGS 4N | 40.65 | 98.38 | 591.3 |

27 | 111280 | IL | CARLINVILLE | 39.29 | 89.87 | 189.3 | 74 | 253735 | NE | HEBRON | 40.18 | 97.59 | 451.1 |

28 | 111436 | IL | CHARLESTON | 39.48 | 88.17 | 198.1 | 75 | 253910 | NE | HOLDREGE | 40.45 | 99.38 | 707.1 |

29 | 112140 | IL | DANVILLE | 40.14 | 87.65 | 170.1 | 76 | 254110 | NE | IMPERIAL | 40.52 | 101.66 | 999.7 |

30 | 112193 | IL | DECATUR WTP | 39.83 | 88.95 | 189 | 77 | 254440 | NE | KIMBALL 2NE | 41.25 | 103.63 | 1,435 |

31 | 112483 | IL | DU QUOIN 4 SE | 37.99 | 89.19 | 128 | 78 | 254900 | NE | LODGEPOLE | 41.15 | 102.64 | 1,168 |

32 | 113335 | IL | GALVA | 41.17 | 90.04 | 246.9 | 79 | 254985 | NE | LOUP CITY | 41.28 | 98.97 | 627.3 |

33 | 113879 | IL | HARRISBURG | 37.74 | 88.52 | 111.3 | 80 | 255080 | NE | MADISON | 41.83 | 97.45 | 481.6 |

34 | 114108 | IL | HILLSBORO | 39.15 | 89.48 | 192 | 81 | 255310 | NE | MC COOK | 40.22 | 100.62 | 796.1 |

35 | 114198 | IL | HOOPESTON 1 NE | 40.47 | 87.66 | 216.4 | 82 | 255470 | NE | MERRIMAN | 42.92 | 101.71 | 986 |

36 | 114442 | IL | JACKSONVILLE 2E | 39.73 | 90.2 | 185.9 | 83 | 255565 | NE | MINDEN | 40.52 | 98.95 | 658.4 |

37 | 114823 | IL | LA HARPE | 40.58 | 90.97 | 210.3 | 84 | 256135 | NE | OAKDALE | 42.07 | 97.97 | 521.2 |

38 | 115079 | IL | LINCOLN | 40.15 | 89.34 | 177.7 | 85 | 256570 | NE | PAWNEE CITY | 40.12 | 96.16 | 378 |

39 | 115326 | IL | MARENGO | 42.29 | 88.65 | 248.4 | 86 | 256970 | NE | PURDUM | 42.07 | 100.25 | 819.9 |

40 | 115712 | IL | MINONK | 40.91 | 89.03 | 228.6 | 87 | 257070 | NE | RED CLOUD | 40.1 | 98.52 | 524.3 |

41 | 115768 | IL | MONMOUTH | 40.92 | 90.64 | 227.1 | 88 | 257515 | NE | SAINT PAUL 4N | 41.27 | 98.47 | 541 |

42 | 115833 | IL | MORRISON | 41.80 | 89.97 | 183.8 | 89 | 257715 | NE | SEWARD | 40.9 | 97.09 | 438.9 |

43 | 115901 | IL | MT CARROLL | 42.1 | 89.98 | 195.1 | 90 | 258395 | NE | SYRACUSE | 40.68 | 96.19 | 335.3 |

44 | 115943 | IL | MT VERNON 3 NE | 38.35 | 88.85 | 149.4 | 91 | 258465 | NE | TECUMSEH 1S | 40.35 | 96.19 | 338.3 |

45 | 116446 | IL | OLNEY 2S | 38.7 | 88.08 | 146.3 | 92 | 258480 | NE | TEKAMAH | 41.79 | 96.23 | 338.3 |

46 | 116526 | IL | OTTAWA 5SW | 41.33 | 88.91 | 160 | 93 | 258915 | NE | WAKEFIELD | 42.27 | 96.86 | 423.7 |

47 | 116558 | IL | PALESTINE | 39 | 87.62 | 140.2 |

**Figure 1.**Illustration of study area and selected meteorological stations. The study area covers three states of the United States: Iowa, Illinois and Nebraska. Stations are marked as circle dots, and colors are labeled for different states. The number of meteorological stations of Iowa (blue dots), Illinois (green dots), and Nebraska (red dots) is 23, 33, and 37, respectively.

**Figure 2.**Distributions of normalized mean Normalized Difference Vegetation Index (NDVI), fractal dimension (fd), and AGDDs along the corn life cycle (Iowa, 2007).

**Figure 4.**NASS’s CPRs Normalization, Iowa (2011). PS = pre-season. PL = planted, EM = emerged, SI = silking, DO = dough, DE = dent, MA = mature, and HA = harvested. (

**a**) original corn progress percentages; (

**b**) normalized corn progress percentages.

## Share and Cite

**MDPI and ACS Style**

Shen, Y.; Wu, L.; Di, L.; Yu, G.; Tang, H.; Yu, G.; Shao, Y.
Hidden Markov Models for Real-Time Estimation of Corn Progress Stages Using MODIS and Meteorological Data. *Remote Sens.* **2013**, *5*, 1734-1753.
https://doi.org/10.3390/rs5041734

**AMA Style**

Shen Y, Wu L, Di L, Yu G, Tang H, Yu G, Shao Y.
Hidden Markov Models for Real-Time Estimation of Corn Progress Stages Using MODIS and Meteorological Data. *Remote Sensing*. 2013; 5(4):1734-1753.
https://doi.org/10.3390/rs5041734

**Chicago/Turabian Style**

Shen, Yonglin, Lixin Wu, Liping Di, Genong Yu, Hong Tang, Guoxian Yu, and Yuanzheng Shao.
2013. "Hidden Markov Models for Real-Time Estimation of Corn Progress Stages Using MODIS and Meteorological Data" *Remote Sensing* 5, no. 4: 1734-1753.
https://doi.org/10.3390/rs5041734