Beaufort+Wind+Scale

=**__Beaufort Wind Scale__**=

Introduction
The Beaufort scale is an empirical measure for describing wind intensity based mainly on observed sea conditions, e.g. how high the waves are. It is based on a scale that ranges from 0-12. It was created in 1805 by an irishman called Sir Francis Beaufort, who was a British naval officer. At that time, it was the naval officers who were in charge of the weather reports, but there was no set scale, so the result was very subjective. A calm day for one man could mean a stiff breeze for another. The initial scale from zero to twelve did not reference wind speed numbers, but the effects on the sails of a ship, from "just sufficient to give steerage" to "that which no canvas could withstand". In 1906, the discriptions was changed to how the sea behaved, instead of the sails, and also land observations were made.The Beaufort scale was extended in 1946, when Forces 13 to 17 were added. However, Forces 13 to 17 were intended to apply only to special cases, such as tropical cyclones. Nowadays, the extended scale is only used in Taiwan and mainland China, which are often affected by typhoons. Wind speed on the 1946 Beaufort scale is defined by the empirical formula: //v// = 0.836 //B//3/2n/s where //v// is the equivalent wind speed at 10 metres above the surface and //B// is Beaufort scale number. For example, //B// = 9.5 is related to 24.5 m/s which is equal to the lower limit of "10 Beaufort". Here is a rough idea of how the scale works Today, hurricanes are sometimes described as Beaufort scale 12 through 16, very roughly related to the standard [|Saffir-Simpson Hurricane Scale] where Category 1 is equivalent to Beaufort 12. However, the Saffir-Simpson Scale does not match the extended Beaufort numbers above 13. Category 1 tornadoes on the [|Fujita] and [|TORRO] scales also begin roughly at the end of level 12 of the Beaufort scale but are indeed independent scales.
 * Beaufort number (force) |||| Wind Speed || Wave height (feet) || WMO* description || Effects observed on the sea ||
 * knots || mph ||
 * 0 || under 1 || under 1 || - || **Calm** || Sea is like a mirror ||  ||
 * 1 || 1 - 3 || 1 - 3 || 0.25 || **Light air** || Ripples with appearance of scales; no foam crests ||  ||
 * 2 || 4 - 6 || 4 - 7 || 0.5 - 1 || **Light breeze** || Small wavelets; crests of glassy appearance, not breaking ||  ||
 * 3 || 7 - 10 || 8 - 12 || 2 - 3 || **Gentle breeze** || Large wavelets; crests begin to break; scattered whitecaps ||  ||
 * 4 || 11-16 || 13-18 || 3½ - 5 || **Moderate breeze** || Small waves, becoming longer; numerous whitecaps ||  ||
 * 5 || 17-21 || 19-24 || 6 - 8 || **Fresh breeze** || Moderate waves, taking longer form; many whitecaps; some spray ||  ||
 * 6 || 22-27 || 25-31 || 9½-13 || **Strong breeze** || Larger waves forming; whitecaps everywhere; more spray ||  ||
 * 7 || 28-33 || 32-38 || 13½-19 || **Near gale** || Sea heaps up; white foam from breaking waves begins to be blown in streaks ||  ||
 * 8 || 34-40 || 39-46 || 18-25 || **Gale** || Moderately high waves of greater length; edges of crests begin to break into spindrift; foam is blown in well-marked streaks ||  ||
 * 9 || 41-47 || 47-54 || 23-32 || **Strong gale** || High waves; sea begins to roll; dense streaks of foam; spray may begin to reduce visibility ||  ||
 * 10 || 48-55 || 55-63 || 29-41 || **Storm** || Very high waves with overhanging crests; sea takes white appearance as foam is blown in very dense streaks; rolling is heavy and visibility is reduced ||  ||
 * 11 || 56-63 || 64-72 || 37-52 || **Violent storm** || Exceptionally high waves; sea covered with white foam patches; visibility further reduced ||  ||
 * 12 || 64 and over || 73 and over || 45 and over || **Hurricane** || Air filled with foam; sea completely white with driving spray; visibility greatly reduced ||  ||
 * * World Meteorological Organization ||
 * * World Meteorological Organization ||

__Did you know?__ The highest wave height recorded in history was 530 meters high. It hit an island and covered it. You can read more on: [|www.surfersvillage.com/surfing/25499/news.htm]

**Purpose**
The goal of our project is to create a model that fits the data and show a relationship between the Beaufort number and the height of the wave. On a first glance at the data, it is obvious that as the wind speed increases, the wave height and beaufort number increase as well. In this page, we will find a more specific type of relationship between the beaufort number and the wave height. We will be following a set criteria and instructions that will lead us to discovering the optimum type of model that would fit this data set.

**Data**

 * Beaufort number || Wind speed (km/hr) || Wave height (m) ||
 * 0 || <1 || 0 ||
 * 1 || 2-5 || 0.15 ||
 * 2 || 6-11 || 0.3 ||
 * 3 || 12-20 || 0.6 ||
 * 4 || 21-29 || 1.6 ||
 * 5 || 30-39 || 3.1 ||
 * 6 || 40-50 || 4.7 ||
 * 7 || 51-61 || 6.2 ||
 * 8 || 62-74 || 7.8 ||
 * 9 || 75-87 || 9.3 ||
 * 10 || 88-101 || 10.8 ||

The wind speed is only for reference.
 * The x values are the Beaufort number and the y values are the wave height.

Question: Why haven't Beaufort number 11 and 12 been recorded? Answer: We think this is because in conditions like 11+, the weather becomes too dangerous to be able to measure safely the wave height of a body of water, so specific data could not be obtained accurately.


 * Graph with original data:

The points form a curve and looks like some kind of exponential function, but we cannot determine the line directly by this curve. So we are going to linearize the data and then 'log' the equation to fit the data.**

=**Linearizing the Data**=

On Fathom, we created three separate plots - (logx, y) (x, logy) and (logx, logy)


 * Logx, y**

The points still resemble some sort of curve. So we tried a different way using (x, logy)


 * x, Logy**

The points on this (x,logy) graph still resemble a curve. So we tried using (logx,logy)


 * Logx, Logy**

It seems as if the (Logx, Logy) graph is the most linear. We went ahead and created a least-squares line of best fit that had a respectable r^2 or 0.97.

Determining the most appropriate type of function

 * //A - Perhaps You Have an Exponential Model//:** **(f(t) = ab^t)**

To find an exponential model to fit the data set in two ways:

1. "By Hand"**-** (Because in an exponential equation there cannot be a zero as one piece of data, since anything muliplied by zero would equal zero, we have decided to take away the 'zero' factor from the first row.) The common ratio -- //b// The constant multiplier -- //a// This should be the first data set on the table, but since this would be a zero, it is impossible to multiply by zero to get a varied number. Therefore the constant multiplier should be **0.15** //Note: Because we have moved the table up by one (as we have taken away the first row), it is only reasonable// //that the 't' factor should move one over too, so instead of the power of t, we will have the power of (t-1)// Residual plot for this equation: Sum of residuals: 6.504
 * || **Data No.** || **Data** || **Ratio from Last Data** ||
 * || 1 || 0.15 || N/A ||
 * || 2 || 0.3 || 2 ||
 * || 3 || 0.6 || 2 ||
 * || 4 || 1.6 || 2.666666667 ||
 * || 5 || 3.1 || 1.9375 ||
 * || 6 || 4.7 || 1.516129032 ||
 * || 7 || 6.2 || 1.319148936 ||
 * || 8 || 7.8 || 1.258064516 ||
 * || 9 || 9.3 || 1.192307692 ||
 * || 10 || 10.8 || 1.161290323 ||
 * |||| **Average Ratio** || **1.672345241** ||
 * Equation by Hand: f(t) = 0.15 * 1.672^(t-1)**

2. ExpReg on calculator: a = .1558980024 b = 1.622280748 r^2 = .9219142692 r = .960136679
 * Equation by Calculator: f(t) = 0.1559 * 1.6223^t**

Residual plot for this equation: Sum of residuals: -6.357

Both of these equations concide with each other fairly well, judging by the closeness of the two lines (as seen below). But referring to the residual plots and the sum of the residuals, it is apparent that the exponential regression on calculater peformed a more accurate process to obtain the formula. Looking at each residual plot, each follows a similar pattern, as the points gradually move upwards, away from the trend line, and then a sudden drop, which agan the points move away from the line. We don't think the exponential model is a good way to describe the data, since we can only imagine the points get furthur and furthur away from the line, the last point is especially quizzical, as it is quite far away from the trend line. The points are not evenly distributed, and the sum of the residuals reflect this, both numbers, 6.5, and -6.3 are far from the expected 0.
 * Analysis**

Graph showing both exponential equations plus data:

To test the equation, we found the wave height with Beauford Scale 11 and 12 Equation 1: f(t) = 0.15 * 1.672^(t-1) t = 11 f(t) = 25.751 t = 12 f(t) = 43.079 Equation 2: f(t) =f(t)= 0.1559 * 1.6223^t t = 11 f(t) = 31.932 t = 12 f(t) = 51.802
 * Extrapolation**

We do not think that the exponential model fits the data very well.
 * Conclusion**

//**B - Perhaps You Have a Inverse Function**//

We have decided as a group that since the power function does not seem to work, and that our data set looks nothing like an inverse of an exponential equation, there is no point in finding out whether if it is an inverse function or not. Our group has decided that we will proceed with the next step and returning to section B if nessessary, since this section does not really apply to our data.

//**C - Perhaps You Have a Power Function: (f(x) = ax^b)**//

The least square line from the logx, logy graph is y= 2.05x-1.00 slope =2.05 intercept = -1.00
 * Create an equation**

Calculate the common difference between logy = a + b*log(x) logy=blogx + a logy= 2.05logx - 1.00 d=2.05=slope a= -1.00 = y-intercept.

logy=logx^d + log 10^a logy= logx^d * 10^a y=x^d * 10^a a= intercept d = slope

We start off with the equation for the log graph: logY=2.05logX -1.00
 * Make a log equation/function**

We then remove the logs y=10^(2.05logX -1.00) y=10^(log(x^2.05))/(log1.00) y=(x^2.05)/10

To get the final equation y=0.1*x^2.05

We used PwrReg on our calculator, and we got the exact same equation.

We added our final equation to our original graph.
 * Fitting the equation**



Residual plot Our residual plot looks reliable. There are some points above the line and some below. There are no outlying points or clumps of points in the plot. The points are pretty spread out.

Since both equations, the one done by hand and the one used in Power Regresssion in the calculator are the same, there is no need for a comparison between the two, they both agree. The equation of the power function does seem to fit the graph fairly well. Although in the residual plot, the data points do sort of move up and down, like a wave, the overall range is not too big, with the largest distance from the trend reaching only about 0.8. The sum of the residual is 1.9, so the sum is not that far away from the desired 0. Overall the residual shows that this line fits in much better than the other residual plots, such as the exponential, with a whopping -6.35 and even more.
 * Analysis**

According to our modelled equation, y=0.1*x^2.05 we can make predictions for the height of the wave for larger beaufort numbers. If we want to find out what is the beaufort number if the wave was 50m high we could substitute y=50 into the equation to find x, which is the beaufort number value. Workings: 50=0.1*x^2.05 500=x^2.05 x=20.72 x equals approx 21 Therefore the beaufort number would be 21.
 * Extrapolation**

Let's say we want to find out the wave height if the Beaufort number is 17. Then we would substitute 17 for the x value y=0.1*17^2.05 y=33m That means the height of the wave would be 33m high when the Beaufort number is 17.

Conclusion
The relationship between the beaufort number and the wave height is a power function. Based on the work above, **we think that y=0.1*x^2.05** is the best model for this set of data.**

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