River+Basin+Drainage

= River Basin Drainage = Basin drainage is an area of land where water from rain or snow drain downhill into a body of water, in this case, a river. Here, we are trying to find a relationship between the length of the river and the river basin drainage area.
 * Introduction:**


 * Rivers || Drainage Area || Length ||
 * Amazon || 6915 || 6296 ||
 * Yenisey || 2580 || 4438 ||
 * Congo || 3680 || 4371 ||
 * Mackenzie || 1790 || 4241 ||
 * Mississipi || 2980 || 3778 ||
 * Volga || 1380 || 3678 ||
 * St. Lawrence || 1030 || 3058 ||
 * Rio Grande || 570 || 3034 ||
 * Essequibo || 155 || 970 ||
 * Mamberamo || 79.44 || 650 ||
 * Nashua || 1.954 || 90 ||


 * PART 1**

1. **“By Hand”** In order to determine the equation by hand, the data set was first linearized by (logx, logy) (explained in Part II). For best-fit-line, the Median-Median method was used. The data set was divided into M1, M2, M3 in 4,3,4. The medians of each section were calculated (above) and the slope was determined using the median points of M1 and M3. Then the Y-intercept was calculated The median-median equation, in its linear form is y= 0.50441x + 1.9075 this was then put back into a power function.
 * **M1** ||  || **M2** ||   || **M3** ||   ||
 * x || y || x || y || x || y ||
 * 3.8396 || 3.7991 || 3.2529 || 3.6275 || 2.7559 || 3.4820 ||
 * 3.5658 || 3.4742 || 3.1399 || 3.5667 || 2.1903 || 2.9868 ||
 * 3.4742 || 3.6406 || 3.0128 || 3.4854 || 1.9000 || 2.8129 ||
 * 3.4116 || 3.5773 ||  ||   || 0.2909 || 1.9542 ||
 * (3.5200, 3.6439) ||  || (3.1399, 3.5667) ||   || (2.0452, 2.9000) ||   ||

2. **Using calculator** - PwrReg y=a*x^b a=70.293 b=0.52745 r^2=0.976 r=0.988 so the equation is: y=70.293x^0.52745

Below is the graph of the data with the lines calculated both by hand and calculator, as well as the residual plot of the equation from the calculator. The blue line indicates calculator's line while the brown one is the line calculated by hand. and below is the residual plot for the model calculated by hand: The equation determined by "hand" is y=80.816x^0.50441 and the equation determined by the calculator is y=70.291x^0.52745 When looking at the two equations, there seems to be a big difference in the y-intercepts and not as much in the slopes, since the y-intercepts are different by about 10, while the slopes are different by about 0.02. However when looking at the graphs, the difference in the two lines are mostly the slopes, and not where it started. As the x values get larger, the effect the slope has on the y value gets bigger. Therefore, the two equations are quite different. At first when looking at the correlation coefficient of PwrReg, which was 98.8%, the equation determined by the calculator seemed better. However, once we looked at the residual plots of the two, we saw that the equation determined by hand was better. Although both of the residual plots were scattered evenly on both sides of the x axis, the residual plot of "by hand" were closer to the x axis overall.
 * Comparison between the two equations, which one is better?:**

The equation obtained calculates the river basin drainage area in relation to the length of the river. The equation was determined through looking at different rivers around the world, such as Amazon, Congo, Mississippi, etc. rivers. We decided to determine the drainage area of other rivers not mentioned in the data. We calculated the drainage area of Niger river and Yangtze river, then compared the values with the actual areas. x=the drainage area (in thousand km^2) and y=the length (km).
 * Extrapolation:**

Niger River, length - 4180km y=80.816x^0.50441 4180=80.816x^0.50441 51.72=x^0.50441 x=2,497

Yangtze River, length - 6211km 6211=80.816x^0.50441 76.85=x^0.50441 x=5475

The calculated drainage area of Niger river, 2,497,000km^2, is quite accurate since the actual area is 2,261,763km^2. On the other hand, the calculated drainage area of Yangtze river, 5,475,000km^2 was completely off from the actual area of 1,800,000km^2. Ideally, the Yangtze river should have larger drainage area, if comparing with that of Amazon river, which was in the data set that was used for determining the equation. This shows that drainage area of rivers vary and that it is hard to come up with a model that agrees with all the rivers in the world.

we tried (logx, y) we tried (x, logy) we tried (logx, logy) and it was the best linearization Using LinReg on the calculator logy=0.5271logx+1.85 log y=log x^0.5271+1.85 10 ^ (logy) = 10^(logx^0.5271)X10^(1.85) y=70.291x^0.52745
 * PARTTWO:Linearizing data :)**