Start a page for count models, work on #298

topics/Count_Models.md

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+---
+title: "Count_Models"
+categories:
+  - Statistical Methods
+---
+
+This page discusses why count models are necessary in certain applications, and
+discusses beginning details of the Poisson, negative binomial, and hurdle models.
+
+## Continuous versus count outcomes
+Typical [regression models](Regression_Analysis) are aimed at predicting the
+response of an outcome variable $y$ to a series of input variables $X = [x_1, x_2, \ldots x_p]$.
+The result is a linear equation of a vector $\beta$ that describes the relationship
+between each element of $X$ and the outcome $y$.
+
+$$y = X\beta$$
+This regression framework assumes that $y$ is a continuous variable, meaning
+that it can take any numeric value within a particular range. The plot below
+shows the relationship between the distance between home and workplace on the
+$x$ axis, and the total miles driven on all vehicle trips on the $y$ axis,
+for a sample of 5,000 reported car commuters who responded to the 2017 NHTS.
+Both of these variables are continuous, meaning that a simple $y = X\beta$
+regression model is appropriate, though more information might need to be
+added to the model below to improve its fit and help explain outlying observations
+or control for heteroskedasticity.
+
+
+![](count_continuous-1.png)<!-- -->
+
+But consider the plot below, showing the distance between home and work on the $x$ axis and
+the number of vehicles owned by the commuter's household on the $y$ axis. Because
+the number of vehicles is discrete and not continuous, the plot looks kind of
+funny. But more importantly than this, we want a model that will predict
+a discrete number of vehicles as an outcome variable, and the blue regression
+line we estimated below will predict between 2.5 and 3.2 vehicles per household;
+this isn't ideal.
+
+
+![](count_discrete-1.png)<!-- -->
+
+## Poisson Model
+
+
+### Negative Binomial Model
+
+
+## Hurdle Model