Predicting the Trajectory of Atlantic Hurricanes with a Statistical Model

Thomas Varley • José Koluder-Ramírez • Lucas Pham

Abstract

A hurricane is a tropical cyclone with maximum wind speeds greater than 74 miles per

created. A multiple linear regression model, a regression-based decision tree, and an XGBOOST

model were trained to compare different approaches to predicting hurricane trajectory. The input

of each model was the two location metrics, longitude and latitude, or the two distance metrics,

distance and bearing, as well as two intensity metrics which were maximum wind speed and

maximum wind pressure. An important hyperparameter chosen was how many previous six-hour

timestamps would be included to predict the location of the next timestamp. Using differing

number of timestamps, the optimal statistical models created consisted of the multiple linear

regression model with four six-hour timestamps used to predict the next six-hour location and

intensity of the storm. It was determined that normalizing this model increased the accuracy of

the results from this regression model. The multiple linear regression was accurate using its own

exceeding 74 miles per hour (8). After forming over the Atlantic Ocean, hurricanes typically

travel north, where they make landfall over North America or the Carribean. When these

hurricanes reach land, the intense wind and rainfall can be catastrophic, often spurring severe

economic instability, claiming the lives of numerous individuals, and causing irreparable damage

to the environment through the destruction of infrastructure that stores toxic materials (7). As

such, it is incredibly important to be able to accurately predict where a hurricane will travel after

its formation, so that people can evacuate the likely path and precautions can be taken to protect

any infrastructure that would be in danger.

The National Weather Service (NWS) and the National Hurricane Center (NHC) have a

variety of models that can be utilized in order to predict the future path of hurricanes. These

models generally fall into one of two categories: dynamical models or statistical models (2).

Dynamical models are incredibly complex, as they account for the physical equations of motion

in the atmosphere and utilize this information to predict the future path of the hurricane. While

dynamical models are oftentimes the most accurate predictor of a hurricane path, their

complexity can lead to exceptionally long runtimes in generating those predictors (1). Statistical

models are much simpler and much faster than their dynamical counterpart, as they utilize

statistical formulas to generate their predictions through the use of historical hurricane data (2).

a linear regression model, an XGBOOST model, and a regression-based decision tree. For each

of the three models, an input consists of a series of location and intensity metrics, which the

models then used to generate their prediction for the path of the hurricane. For the location

metrics, longitude and latitude of the hurricanes were applied first to each of the three models to

generate predictions and then switched the location metrics to distance and bearing in order to

compare and determine which predictions were more accurate. For the intensity metrics, the

maximum wind speed and maximum wind pressure were applied to each of the three models.

Beyond simply altering which metrics were input for each model, it was also necessary to

select the optimal number of six-hour timestamps would be utilized by each statistical model to

predict the next location of a hurricane. This number of timestamps served as an important

normalization, this model was able to further reduce mean absolute error and improve in

predictive accuracy. This result demonstrates that a more complex model does not necessarily

lead to a more accurate model, as multiple linear regression models that utilized more than four

six-hour timestamps to make their predictions often suffered from overfitting and were more

inaccurate as a result.

Methods

The data from the National Hurricane Center came in the form of a text file. This given

data is a (50911 x 23) matrix with 50911 observations and 23 different variables describe 1864

total hurricanes that occurred in the Atlantic Ocean from 1851 to 2018. The file was not a CSV

and could not be read traditionally with built-in Pandas functions. For each hurricane, the

identifying information was listed such as hurricane name, year, and unique identifying codes.

Then there was a variable amount of data for each six-hour timestamp provided. Each timestamp

consisted of the time and date and the atmospheric measurements. A PDF was provided that gave

a detailed explanation of how to successfully read the text file. To acquire the data, each line of

the file was read and relevant information was extracted using the specific indexing information

To create a model, it was determined what features would be relevant. It was then

decided to include the latitude and longitude which represent the location of the hurricane at a

timestamp as well as the maximum wind speed and pressure to represent the intensity of a

hurricane at a timestamp. The other features included in the data provided by the National

Hurricane Center were nearly all missing for hurricanes before 2004 which represented a

significant portion of the data. Thus, those features were not used in the models. To determine

the optimal number of timestamps included, models were tested using differing numbers of

timestamps from 3 timestamps to 10 timestamps of all 4 input variables. Thus, the input size

ranged from (3,4) to (10,4). To organize the list of hurricane objects obtained by extracting the

data from the text file into input and output data, hurricane objects that had fewer observations

able to provide 5 input and output pairs using a window length of 5 where each input had

dimension (5,4) and each output had dimension (1,4). As all models required a two-dimensional

input and the input was three dimensional once all training samples were combined, the input

matrix was transposed so that a single input would have a dimension of (1, window length * 4).

Additionally, as all training inputs were created, a list of the actual lines used to make the input

and output respectively without repeats. For example, a hurricane with only 10 timestamps

would contribute the first 9 timestamps to this non-repeated list.

Three different variations using all three types of models were first investigated. Using all

three models with the data as is, normalizing the data, and determining the distance and bearing

from the latitude and longitude between two points to replace the latitude and longitude data

points as in the input were three variations of our original data that were hypothesised to have an

impact on the performance of the model. Once the data has been organized into training input,

training output, test inputs, test outputs and list of training lines without repeats one could easily

use the training inputs and outputs to train the model and the test inputs and outputs to evaluate

the model’s performance by mean absolute error. To see the effect of normalizing the data, the

Python library StandardScalar was used. A scalar was trained on the list of training inputs

without repeats and a scalar on the test inputs to determine the mean and standard deviation of

and bearing as opposed to the longitude and latitude, it was determined how to calculate the

distance on the Earth in terms of kilometers between two points using a builtin geopy function

and using a bearing value estimated from the two coordinates. To convert an input of a certain

length with longitude and latitude values to distance and bearing values, the calculation utilized

the distance and bearing between consecutive timestamp locations. Once the distance and

bearing are calculated the later timestamp’s maximum wind speed and maximum wind pressure

were used as the intensity metric. Thus, an input consisting of 5 timestamps with longitude and

latitude coordinates will create an input of 4 timestamps or, in general, one less timestamp than

the original input’s window length. The output will be the distance and bearing from the original

input’s last timestamp location to the original output’s timestamp location. The output’s

a model’s accuracy in determining a given number of timestamps in the future, a prediction was

made using the first window length of inputs. Then all inputs except the first time stamp were

used in addition to the new prediction to predict the second timestamp and so on until we

reached the desired amount of timestamps in the future. Once we reach that desired timestamp,

we recorded the prediction and the actual values for the output. Using all the test hurricanes that

had the required number of timestamps to predict the necessary timestamps into the future, we

recorded the mean absolute error for a given number of timestamps in the future.

Standard Score Normalization (or Z-scores) linearly transforms data values to having a mean of

zero and a standard deviation of 1. The Z-score of each value is equal to the differenceZ

used to normalize longitude, latitude, wind speed, and wind pressure.

Mean Absolute Error

In this paper, Mean Absolute Error (MAE) is used to justify the correctness of predictions for

different window lengths instead of the normally used Mean Square Error (MSE) or Root Mean

Square Error (RMSE). In this case, are the differences between the actual values in they

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justification tool than RMSE for this analysis. First of all, between the two, from an

interpretation standpoint, MAE is beneficial due to its simplicity. Secondly, since the errors in

this model are considerably small, it’s more appropriate to use MAE instead of RMSE, since

RMSE tends to work more effectively with penalizing large errors (5).

Multiple Linear Regression

Multiple Linear Regression is being used in this project as one of the three main fitting models

(along with Decision Tree and XGBOOST). Multiple Linear Regression is a straightforward

method and that can be interpreted to assess what features are most important in predicting

tan2(X, Y ) a

between the positive Y-axis and the ray to the point , while X and Y are two

X, Y ) ≠ (0, 0) (

quantities and can be calculated as:

B

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A

c

B

c

Decision Tree Regression

When it comes to small-to-medium structured/tabular data, decision tree based algorithms are

considered best-in-class as of the time of this report (3). Therefore, this method of statistical

modeling is used as one out of three models to train the data. Decision tree learning uses a

decision tree (as a predictive model) to go from observations about an item (represented in the

branches) to conclusions about the item's target value (represented in the leaves). Based on

specific conditions, the tree is being splitted to more branches and creates more sub-trees as the

tree’s height increases. The main reason why both Decision Tree Regression and Multiple Linear

boosting framework implemented in Python. Moreover, XGBoost is described as the top

tree-based algorithm, integrated from Decision Tree, that optimized Gradient Boosting algorithm

through parallel processing, tree-pruning, handling missing values and regulations to avoid

overfitting and bias (6). Although XGBoost doesn’t work best for all given prediction models,

it’s such a powerful tool that should always be considered. Furthermore, since Decision Tree is

one of the three models we decided to use, XGBoost can serve as a tool to compare the gap in

predicting performance between the two.

Figure 2: Scatter plots depicting MAE per Window Length using various models

Top Left: Scatter plot depicting the mean absolute error of location features using all three

models across different window lengths. Top Right: Scatter plot depicting the mean absolute

error of all features using all three models across different window lengths. Middle Left:Scatter

plot depicting the mean absolute error of location features using all three models across

different window lengths with normalized input. Middle Right:Scatter plot depicting the mean

absolute error of all features using all three models across different window lengths with

normalized input. Bottom Left: Scatter plot depicting the mean absolute error of distance

features using all three models across different window lengths with distance and bearing.

Bottom Right: Scatter plot depicting the mean absolute error of all features using all three

models across different window lengths with distance and bearing.

The first objective of this investigation was to determine the optimal model for hurricane

and the information would be lost by normalizing the angle. The distance based metrics may

have performed poorly because converting the variables to distance between location variables

could potentially result in a loss of information as to where the hurricane is actually at during a

timestamp. For example, hurricanes in the Gulf of Mexico may generally behave differently than

hurricanes in the middle of the Atlantic Ocean. That information is preserved when the longitude

and latitude are used, but when distance and bearing are used, the information about the specific

location of the hurricane is lost. Thus, there are significant errors in the models that used distance

and bearing instead of longitude and latitude.

In every graph in Figure 1 where longitude and latitude are the location metrics, the

multiple linear regression produces an output with the least mean absolute error for both

longitude latitude, maximum wind speed, and maximum pressure, there is a locally optimal

window length at four and nine timestamps. In practice, using only four timestamps, or 24 hours,

of data may be more accessible and accurate than determining using nine timestamps, or 54

hours of data. Additionally a model that uses four timestamps is significantly less complex than a

model that uses nine timestamps. The four-timestamp model must only learn 17 parameters

while the nine-timestamp model must learn 37 parameters. Normalizing this model when using

longitude and latitude brought a small decrease in the mean squared error for nearly all window

length. Thus, the optimal model is the normalized multiple linear regression model that uses a

window length of four-timestamps.