# Transportation Deployment Casebook/United States Railroads
## Introduction

The mode choice for analyzing the historic life-cycle of a transportation technology was railroads. Specifically in the United States, the first passenger and freight railroads emerged in the early part of the 19th century in the northeastern territory. Older tracks have been laid early than the 1830s but its main intent was used for transporting coal out the deep mines that people worked in. The original use of railroad tracks was for increasing coal production. When coal was discovered in Wyoming in 1843, this also lead to an angst to increase the distances of lines extending from the Missouri River. Another drive to push west, was the California Gold Rush during the 1850s. People flooded to the west coast to get a piece of the pie and yet again can contribute to the fact that private as well as public personal wanted railroads in that western direction.

## Methodology

In order to analyze the data that was collected from the 1949 Historical Statistics Reference, a three-parameter logistic function was considered. Basically using and fitting an S-curve to the suggested data ultimately will provide a general picture of the transportation technology's historic life cycle of birth, growth, maturity and decline. The logistic function that was utilized was $S(t)=K/\left(1+e^{-b(t-t_{0})}\right)$, where S(t) is the status measure, in this case, kilometers of operated railroad tracks in the United States, t is time which is in years, t$_0$ is the inflection time which is the year in which .5*K us achieved, K being the saturation status level and b is a coefficient. K and b are to be estimated as well.
Figures 1 and 2 show the results of the curve fitting process and the subsequent tables show the summed up results. This methodology appears to have done an sufficient job to create the life cycle of the tracks operated in the US. A single variable linear regression $Y=b*X+c$, which estimates the parameters b and c. The independent variables in this case would be the years associated with the different lengths of railroad track , X, while the dependent variable Y, comes from the function $Y=\ln \left({\frac {OperatedKilometers(t)}{K-OperatedKilometers(t)}}\right)$.
To be satisfied with the results of this entire process, an R-squared value as close to 1.0 one can achieve indicates success, within reason of course. Along with the R-squared value being as close to 1.0 as possible, a value of 2 or greater for the t-statistic is esired as well since that indicates that the estimated variables are statistically significant at a a 95% confidences interval.

## Results
### Table 1: Computed results for the different estimated saturation status (K) levels.

| Trial | K value | Intercept | b value | R$^2$ | Inflection Year t$_0$ |
| 1 | 700,000 | -172.9 | 0.09122 | 0.94020 | 1895.14 |
| 2 | 720,000 | -162.6 | 0.08570 | 0.93661 | 1897.78 |
| 3 | 740,000 | -156.6 | 0.08244 | 0.92966 | 1899.78 |
| 4 | 760,000 | -152.2 | 0.08007 | 0.92306 | 1901.25 |
| 5 | 780,000 | -148.8 | 0.07822 | 0.91710 | 1902.63 |
| 6 | 800,000 | -146.0 | 0.07670 | 0.91174 | 1903.86 |
| 7 | 820,000 | -143.7 | 0.07543 | 0.90691 | 1905.00 |
| 8 | 840,000 | -141.7 | 0.07434 | 0.90253 | 1906.04 |
| 9 | 860,000 | -139.9 | 0.07338 | 0.89854 | 1907.02 |
| 10 | 880,000 | -138.4 | 0.07254 | 0.89488 | 1907.94 |
| 11 | 900,000 | -137.0 | 0.07179 | 0.89152 | 1908.81 |
| 12 | 920,000 | -135.8 | 0.07111 | 0.88842 | 1909.64 |
| 13 | 940,000 | -134.7 | 0.07050 | 0.88554 | 1910.42 |
| 14 | 960,000 | -133.7 | 0.06994 | 0.88286 | 1911.17 |
| 15 | 980,000 | -132.7 | 0.06942 | 0.88037 | 1911.89 |
| 16 | 1,000,000 | -131.9 | 0.06895 | 0.87803 | 1912.59 |
| 17 | 2,000,000 | -117.8 | 0.06099 | 0.83338 | 1931.87 |
| 18 | 5,000,000 | -113.1 | 0.05794 | 0.81367 | 1952.03 |
| 19 | 10,000,000 | -112.2 | 0.05708 | 0.80783 | 1965.75 |

### Table 2: Best value of K chosen due to the highest value of R$^2$

| Variable | Value |
| Saturation Status Level, K | 700,000 |
| Coefficient, b | 0.09122 |
| Inflection Year, t$_0$ | 1895.14 |
| R$^2$ | 0.94020 |

## Analysis

After the data collection and regression processes, the results obtained seemed to fit based on the calculated value of R-squared to be equal to 0.94020 which is illustrated in Table 2. On the other side of the analysis point of view, 'why the shape of the curve is such, can be attributed to politicians during the ladder half of the 19th century. President Abraham Lincoln passed the Pacific Railway Act in 1862 which aimed to promote construction of a railroad and telegraph Line from the Missouri River to the Pacific Ocean. At this point in history, it can be argued that this was the beginning of the rapid growth phase of railroads in the United States. At about the same time, there were four men who became heavily invested in the railroads extending to the west coast. Collis Huntington, Mark Hopkins, Leland Stanford and Charles Crocker, later to be dubbed "The Big Four" invested $1500 each into a new railroad that they saw as as being a very lucrative opportunity. The four men were the board of directors for the newly found Central Pacific Railroad in 1963.
With the data that was collected, Figure 2 shows that there was an approximate 15 year period of maturity which followed an explosive 45 year growth period. It can be speculated that at about 1920 there became a saturation point in which not many new railroads were made. You begin to see a decline in the total length of tracks being operated especially when the Great Depression hit. Since more and more railroad companies were going bankrupt during this time, it made no sense whatsoever to keep line operational.

## References

* ↑ An Act to aid in the construction of a railroad and telegraph line from the Missouri river to the Pacific ocean, and to secure to the government the use of the same for postal, military, and other purposes 12 Stat. 489, July 1, 1862
* ↑ Bureau of the census. U.S. Census, (1949). Historical statistics of the United States. Retrieved from website: http://www2.census.gov/prod2/statcomp/documents/HistoricalStatisticsoftheUnitedStates1789-1945.pdf