Statistical Learning

Supervised vs. Unsupervised Learning

The goal of supervised learning is to fit a model that describes the relationship between a response (dependent) variable and predictors (independent variables). Examples include linear and logistic regression.
In unsupervised learning problems there is no response variable. Goal is to understand relationships between variables or between observations.

Subset of community data. Full data set includes 1,441 observations/samples and 330,409 variables/OTUs.
SampleID	Otu000001	Otu000002	Otu000003	Otu000004	Otu000005	Otu000006	Otu000007
10_1_72	3132	2103	379	82	131	725	74
10_1_73	3090	3913	528	298	463	3093	967
10_1_74	3570	3860	571	267	534	3212	1096
10_1_75	2198	2457	261	192	267	2158	864
10_1_76	2150	1559	326	74	159	494	63
10_1_77	2085	1737	204	107	387	744	44
10_1_78	742	489	75	20	17	255	44
10_1_79	4527	2186	362	138	114	410	78
10_1_80	570	765	37	19	9	176	0
10_1_81	2166	1546	378	96	182	621	153
10_1_82	4050	5974	1178	571	670	3922	214
10_1_83	750	661	84	47	49	480	497
10_1_87	4990	4411	427	231	249	1325	149
10_1_88	2212	3935	246	177	300	1812	225
10_1_89	2022	1840	376	146	553	948	332

This is an example of high-dimensional data
It is difficult to understand how samples relate to each other in 330,409-dimensional space
There are several approaches to finding lower-dimensional representations of data
We will consider a 2-dimensional representation of the community data
Dimension reduction is an example of unsupervised learning

NMDS plot of Macatawa community data. Point colors indicate site type. Shapes indicate water temperature season.

Supervised learning problems can be broken down into two types:

Fit/train a model by producing a set of training data
The training set includes a large number ( \(\approx\) 500,000) of manually labeled (“dead”, “grass”, or “sand”) pixels

Labeled pixels

Trained model can predict pixel type (“dead”, “grass”, or “sand”) for the entire image
In this case, we used an XGBoost model, which uses an ensemble of decision trees

How do we use the classified images to map surface composition over an entire dune complex?
Resolution of maps is approximately 33cm
Stitch about 1,000 multispectral drone images into an orthomosaic
Break this image into 33cm \(\times\) 33cm squares
Predictors are average R, G, B, NIR, RE, NDVI for square
Response is proportion of square covered by grass
Regression problem (response is quantitative)

Training data for regression model.

Map generated using model to predict surface composition of 30 cm \(\times\) 30 cm squares.

In the two dune surface problems we are only interested in prediction
- Predicting type of pixel (“dead”, “grass”, or “sand”)
- Predicting proportion of surface coverage for each class
Otherwise, the relationship between the predictors and the response is not of much interest
In some problems, this relationship is of primary interest
In inference problems, we use the training data to infer relationships between the variables in a population or process

Want to understand factors that influence survival of pioneer plants in cloud forest of Monteverde, Costa Rica
The pioneer plants we are interested in recruit into gaps in the forest canopy
Predictors are plant characteristics (e.g., height, diameter, age) and gap characteristics (e.g., gap size)
Response is whether plant survives or not

term	estimate	std.error	statistic	p.value
(Intercept)	-3.40	0.78	-4.4	<0.001
age	-1.60	0.58	-2.7	0.006
gap_Small	2.24	0.82	2.7	0.007
Ht	-4.60	1.80	-2.3	0.022
DBH	4.10	1.20	3.3	0.001
gap_Small:Ht	3.70	1.90	2.0	0.044
gap_Small:DBH	-2.90	1.40	-2.1	0.034

Logistic model fit to the data
Can use the model to make inferences
For example, as diameter increases, so does probability of survival, but the rate of increase is smaller in small gaps

More complicated/flexible models tend to result in more accurate predictions
However, more complicated models are more difficult to interpret
Sometimes we sacrifice accuracy for interpretability, especially when we are most interested in inference