From f94ab48f80bd407506662a6b5f6874fb825158f3 Mon Sep 17 00:00:00 2001 From: Greg Macfarlane Date: Mon, 14 Jun 2021 17:14:15 -0600 Subject: [PATCH] Add page on count models --- .../topics/countmodels_continuous-1.png | Bin 0 -> 76320 bytes .../public/topics/countmodels_discrete-1.png | Bin 0 -> 54483 bytes .../public/topics/countmodels_outcomes.png | Bin 0 -> 37734 bytes .../topics/countmodels_poissondist-1.png | Bin 0 -> 27343 bytes topics/Count_Models.md | 174 ++++++++++++++++-- 5 files changed, 160 insertions(+), 14 deletions(-) create mode 100644 .vuepress/public/topics/countmodels_continuous-1.png create mode 100644 .vuepress/public/topics/countmodels_discrete-1.png create mode 100644 .vuepress/public/topics/countmodels_outcomes.png create mode 100644 .vuepress/public/topics/countmodels_poissondist-1.png diff --git a/.vuepress/public/topics/countmodels_continuous-1.png b/.vuepress/public/topics/countmodels_continuous-1.png new file mode 100644 index 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zTfxwPdPYGyApb;Sw*l2&fdV|@xk><%#l=ecOOV6Qo=<9<-0yp*7BDFCp3iCl3_)xw z`0##;ypfL-i_gQF5f0^uRXV;R5>fjYDa)~udk6L`LqI4jVr|I7*WrumC|}9BE2`)< z-&C@uva*T$o7Hx!7_;gPB>*F47tWT|#C~pwmt;^Te&B8QN>e0OmU^Ns6tmLjmQ8f~ z6k-2Dza?NaNs`eylPU|h8(7w_+m5QJ%!2E}cjF!vP^eX2`)A6Ib!b>71re@)SuBW% z_UTSdYpIq-D8~FubS~}9E8Ot2PshCENm*g}(kiU`_W>6Tc>hq9Dflroef-^{GRg(s z>LPUW+oj0JP{#6Dex8+$JZqh=Y +![Household VMT versus distance to work.](countmodels_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. +But consider the plot below, showing the same $x$ axis but with +the number of home-based work trips produced by each household on the $y$ axis. +Because the number of trips 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 and 1 trips household; this +isn't ideal. -![](count_discrete-1.png) + +![Household work trips versus distance to work.](countmodels_discrete-1.png) + ## Poisson Model +A better option would be to predict a probability that each household will produce +a certain discrete number of trips. One way to do this is with a Poisson +regression model. In this model, an analyst predicts the mean of a Poisson +distribution with a regression equation (instead of a line). The +Poisson distribution is: + +$$P(k | \lambda_i) = \frac{\lambda_i^ke^{-\lambda_i}}{k!}$$ + +where the probability of a discrete outcome $k$ is determined by the mean +$\lambda_i$ of the distribution. The plot below shows how as the mean increases, +the probability of higher outcomes increases. + + + +![Poisson distribution at different levels of mean.](countmodels_poissondist-1.png) + +A Poisson regression model allows attributes of an observation to affect the +value of the mean. So instead of $y = X\beta$ in the linear regression, we now +have $\lambda = X\beta$, and $\lambda$ gets put into the distribution equation +above. In the model below, average work distance decreases the average number of +trips, but more workers and more vehicles increases the average number. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Linear Poisson
(Intercept) -0.106 -0.570***
(-1.625) (-13.865)
avg_workdist -0.010*** -0.006***
(-6.851) (-7.070)
wrkcount 1.168*** 0.665***
(29.000) (28.801)
hhvehcnt 0.147*** 0.089***
(5.626) (5.925)
Num.Obs. 5533 5533
R2 0.185
R2 Adj. 0.185
AIC 19111.6 17871.7
BIC 19144.7 17898.1
Log.Lik. -9550.813 -8931.830
F 419.097
+ +Of course, this is just an average. In a trip-based model, this average for each +household might be sufficient. But you could also simulate a discrete choice +for each person. The plot below shows the probability of a certain number of trips +made by a sample of households, alongside what the predicted Poisson mean was. +Households with a higher predicted mean have a higher probability of making more +trips. + + + +![Poisson regression model outcomes.](countmodels_outcomes.png) + + ### Negative Binomial Model +The Poisson model assumes that the mean and standard deviation of the distribution +are the same. This can be a bad assumption, because it forces the distribution to +spread out when the mean is higher. The negative binomial model relaxes this assumption, +and might be useful in some contexts. + +### Hurdle Model +The Poisson and negative binomial models assume the same distribution across +all outcomes; this might not be desirable if the number of zeros is high or +low for some structural reason. For example, owning zero vehicles is very different +from owning one or two. A hurdle model breaks the distribution into two different +components: + + - A binomial model determines the probability of choosing zero versus a positive + number. + - A poisson or negative binomial model (with zero removed) determines the + probability of a specific positive number, conditioned on the previous model. + +## References -## Hurdle Model + 1. [UCLA Stats](https://stats.idre.ucla.edu/r/dae/poisson-regression/)