# Continuous distributions as independent variable in regression

Problem: My research issue concerns logistic regression where each observation is an area, not a simple point. As such, each independent variable ($x_i$ of $\boldsymbol{X}$) is a distribution of measurements over that area. The distribution of $x_i \sim g(\mu,\sigma)$ where $g$ can be any family of distributions. The mean and variance of the distribution can be calculated.

Motivation: This problem is derived from spatial sampling. As an example; $Y_i \in \{0,1\}$ is recorded as $Y_1 = 1$ on three acre parcel of land and $x_1$ representing the topographic slope, is recorded at 100 systematically/regularly sampling locations on that parcel. We want to model $y_i \sim f(\beta0+\beta_1x_1)$. We know the samples of $x_1$ represent a continuous distribution that can take on a variety of distributions. We also know that measurements of $x_1$ are highly spatially correlated. The observations of $Y_i = 0$ can be sampled from any location that is not known to be $Y_i = 1$. That is to say negative observations are pseudo-absence or background samples, not true absences. I am interested in both predicting new areas of $Y_i$ and the inference of parameter estimates

Attempted solutions:

• use each sample of the independent variable $x_i$ as an individual sample observation of $Y_i$. In this case, the spatial correlation of $\boldsymbol{X}$ needs to be assumed and modeled. This is potential solution; in which case, what class of models could do?
• Model each area of $Y_i = 1$ as the mean or mode of the measurements of the independent variable $x_i$. In this case, it would be assumed that the mean/mode is meaningful or representative of the distribution of $x_i$, but typically it is not. I do not think this is a potential solution.
• Treat this as an measurement error model. 1) I cannot assume normality of errors, and 2) the additional measurements are real and meaningful, not just noise. I do not think this is a potential solution.
• Treat this as a multi-level model and pool across the measurements of $x_i$. I believe the problem here is that I will be predicting for new $Y_i$ areas that are not represented in the model fit. This seems to work for parameter inference, but not for prediction.
• As a spatial problem, this could be represented as densities, counts, or presence/absence over areal units and the correlation measured and modeled with standard spatial models. However, the prevalence of occurrence is very low (~ 0.001%) and the measurements of $x_i$ are too variable to be meaningfully aggregated into arbitrary areal units. I do not think this is a potential solution.
• I have used ML approaches of classification (classification trees, SVM, etc...) to reduce the the correlation issue with sampling tricks, but it only helps to mitigate the consequences without addressing the issue.
• Marginal models or Population Averaged models seem like a potential solution, but I have not worked it through as of yet.

Similar questions:

I asked a related, but less direct question a while back, but no answers: Logistic regression on clustered presence observations

This is interesting, but I don't think that answer is totally relevant here. I do not want to treat $\mu_i$ of $g(x_i,\mu_i)$ as a random variable as in the case of measurement error. Additionally, a prior on $\mu_i$ would be difficult to construct. Bayesian regression with independent variable drawn from distribution

Similar question, but assumed normality of the $x_i$ distribution and is unanswered. Logistic Regression with (Normal) Distributions for Independent Variables

Research example: To address a comment below, here is a brief research question that this question could apply to. Note: my overall question is meant to be non-specific to any one research domain; it is a question applicable to any domain.

A potential data set could be the location of long-since vanished ancient Iron Age forts in Germany. This data set may contain 10 to 10,000 fort locations. These are recorded as areas (polygons) that can range from small (0.3 acres) to very large (hundreds of acres). The data set also contains point locations that are known to not have been an Iron Age forts. It is observed that the fort locations may be associated with certain ranges of topographic slope and distance to water. The topographic slope (or any co-variate) of each fort location is measured on a regular grid of say 100 meters, therefore small fort areas will have a few slope measurements and larger fort will have more. To fit a logistic regression to these data, one would use the many slope measurements taken from each of the known fort areas (positive case) and the single slope measurements from the non-fort areas (negative case).

This model can be fit of course, but the parameter estimates will be overly optimistic due to correlation in the positive case observations. Alternatively, the many slope measurements could be aggregated to an average slope, but that suffers from the average not being representative. Another approach is to use random effects for each known site area, which can produce good inference on variance within/between locations and parameter estimates, but I do not believe works well to predict the log-odds of for presence on a new area of the landscape (e.g. does this pattern hold up in Belgium?). This example boils down to how to model multiple measurements of a single positive case when the measurements are no considered measurement noise of repeated measures in the more traditional sense.

Update: Possible Directions: The commenting on this question has been interesting and helpful, but the underlying question remains. Below are links to papers and methods that appear to me to approach the data structure that I am describing here. I do not claim to understand these methods fully, but the description of the problems they are meant to address seem very applicable. Perhaps someone can help shed light on whether that is true.

Distribution Regression As described in works by P´oczos and Szab´o (links below), this is a method that finds $Y = f(P) + \mu$ where $f$ is the unknown regression function, $\mu$ is the random error, and $P$ is an unknown distribution described by samples. In my problem statement at the top, the distribution $P$ is a non-parametric representation of $g(\mu,\sigma)$ as described in the opening paragraph of this quetion. The unknown distribution of covariate $P_i$ is described by samples $\{X_{i,1},\dots,X_{i,n_i}\}$, such that $X_{i,j}\sim P_i$. This seems to match the data structure in my example above where I have many measurements of a single covariate that in total describe an observation of the positive/present condition $Y_i = 1$. [note: "Distribution Regression" seems to be a term that can apply to a few different methods. The authors P´oczos and Szab´o are describing the version that I am referencing. Authors Foresi and Peracchi 1995; Chernozhukov et al. 2013; Koenker et al. 2013, discuss a distribution regression that appear to be a different method.]

See PDFs linked below for more details: http://www.cs.cmu.edu/~aarti/pubs/AISTATS13_BPoczos.pdf

http://people.ee.duke.edu/~lcarin/Esther2.18.2014.pdf

http://jmlr.org/papers/volume17/14-510/14-510.pdf

Thank you for your consideration and please inform me of any errors in concepts or notation.

• Although I cannot make sense of the passage beginning "The observations of Yi=0 can be sampled ...," it sounds like you need a spatial generalized linear model. See the links at cran.r-project.org/web/packages/geoRglm. – whuber Nov 14 '16 at 19:58
• Thank you @whuber. That passage is to say that the negative observations are pseudo-absence, not true absences. In other approaches, this can be conceptualized as background samples. edited to clarify – mr.ecos Nov 14 '16 at 20:26
• @whuber, thanks for the suggestion. I looked over the vignette and from my amiditably limited understanding there are two issues; 1) I need to predict to new areas that are not in the fitted random effects (I could use the mean response or conditional on some some attribute, but that is not desireable); and 2) this package is an implementation of kriging which along with most geostatistical models assumes some degree of continuous distribution of $Y_i = 1$ conditional on trends in $x_i$. That assumption is not very well fit to my data generating mechanism. I greatly appreciate your insight. – mr.ecos Nov 15 '16 at 0:42
• @noumenal. Thank you for your comment. Originally, I did not provide a research question because 1) I did not want to ask a "How do I analyse my data?" question; and 2) I tried to ask a question that distilled away the domain specific issues to get to a more fundamental statistical question. That being, how to represent multiple measurements of a single positive case when they are not assumed to be measurement noise or repeated measures in a more traditional clinical case. I am grateful for any ideas you may share. – mr.ecos Nov 15 '16 at 15:27
• Not sure this would help, but just a thought - using MLE: times.cs.uiuc.edu/course/410/note/mle.pdf – noumenal Nov 30 '16 at 6:39