Lecture 09: Gradient Forest and Genomic Turnover

Correlations between environmental predictors and genetic data

Same climate (predictor) but a different response (SNPs).

Two environmental scatterplots contrast presence/background labels with white, gray and black individual SNP dosages.

Keep the environmental predictors; replace occurrence labels with SNP dosages.

Encoding genotype data into a compact format

Count the alternate copies

Paired chromosome cartoons show reference/reference, reference/alternate and alternate/alternate becoming white zero, gray one and black two. Missing calls use a dashed box and question mark.

Two chromosome copies give a dosage of 0, 1, or 2.

Thinking about filtering/QC

Filter for missingness, sufficiently variable sites, LD, and take a subset so it runs faster.

Four cartoons illustrate eight called samples out of ten, one minor allele among twenty called copies, one highlighted SNP per RAD contig, and a repeatable subset of loci.

Filtering chooses which genetic variation enters the analysis.

Timing: 1.5 minutes. Call rate and MAF are computed among selected samples. For diploid calls, alternate frequency is sum of called dosages divided by twice the called sample count; MAF is the smaller of that frequency and its complement. One_per_contig takes the first eligible SNP within each RAD contig before sampling across eligible contigs; this is not formal LD pruning and would be inappropriate as one-per-chromosome filtering. Changing the filters or subset can change the biological summary. Source: course Lab09 and pyGF README. The call-rate cartoon shows 8 of 10 samples, not the real 36-sample dataset. The minor allele panel shows 1 of 20 called allele copies, exactly 5%. The final dot array symbolizes a subset and is not an exact count of 256. These are thresholds, not additional filtering results.

The gradient forest model

One random forest for each SNP

Three temperature–rainfall plots show the same twelve samples A through L at identical environmental positions. White, gray and black points encode dosages zero, one and two at SNP A, B and C. Each SNP panel feeds its own separate random forest.

Each forest sees the same environments and learns one SNP’s dosage pattern.

Timing: 1 minute. Recall that each tree partitions climate space and predicts a mean response within a leaf. Trees receive bootstrap resamples of the observed rows for their SNP. Missing y values are excluded per SNP rather than imputed. This is a collection of separate single-response forests, whose useful information will be synthesized. Source: pyGF README Method section. Walk through one individual, such as A, across all three panels: its position never changes, while its dosage may change by locus. Then compare SNP A, which mainly tracks temperature, with SNP B, which mainly tracks rainfall, and SNP C, which has a different pattern. All twelve samples, coordinates and dosages are invented; this is a two-predictor cartoon of the 19-predictor lab workflow. Each tree glyph symbolizes a tree within the SNP-specific ensemble, not an additional response. Missing calls are omitted per SNP in the real fit.

"Out of bag" prediction error

Evaluating ability to predict genotype of holdout set of samples

Average predictions from trees that never learned from individual A.

Evaluating model fit

Aggregating OOB prediction error across all Trees/SNPs

Observed genotype dots and prediction crosses compare perfect predictions, constant-mean predictions, and reversed predictions. Their R squared values are one, zero, and minus three.

R² = 0 matches the mean-only benchmark; positive scores improve on it.

Timing: 1 minute. The exact formula is 1 minus sum((y - yhat_OOB)^2) divided by sum((y - mean(y))^2), over rows with OOB predictions. The reference mean is calculated over the scored observations. Constant responses are skipped; sparse responses and inadequate OOB coverage are also diagnostic concerns. Positive R² is a predictive screening rule, not a significance test for adaptation. Source: pyGF forest.py and https://scikit-learn.org/stable/modules/generated/sklearn.metrics.r2_score.html. In all three panels the six observed dosages are 0, 0, 1, 1, 2, 2, whose mean is 1 and total squared variation is 4. Predictions equal to the observations have SSE 0 and R² 1. Constant-one predictions have SSE 4 and R² 0. Reversed predictions have SSE 16 and R² −3. These scores are exact for the toy points, not additional results from pyGF.

The fitted example

Some SNPs contribute; others do not

Histogram of fitted SNP OOB R squared values. Gray marks excluded responses and teal marks responses retained in the turnover summary; the dashed line marks zero.

pyGF retains SNPs with positive OOB R² and positive permutation importance.

Evaluating climate variable importance

Shuffle climate; see what breaks

The same six individuals have unchanged dosage values but shuffled temperature values. A fixed step prediction rule then makes larger errors, increasing mean squared error from zero to 1.33.

If prediction error grows, the fitted forest relied on that variable.

Timing: 1 minute. The forest is not refitted after shuffling. We break part of its predictive information at evaluation time. pyGF stores raw OOB MSE increases in permutation_importances_. This differs from the impurity decrease importance we saw in a simple random forest: both appear in the GF algorithm, but serve different roles. Source: pyGF README Method section; general principle at https://scikit-learn.org/stable/modules/permutation_importance.html. The original temperatures are 5, 7, 9, 11, 13, 15; observed dosages are 0, 0, 1, 1, 2, 2. The fixed toy rule returns 0 below 8, 1 from 8 to below 12, and 2 from 12 onward. The shuffled temperatures yield squared errors totaling 8 across 6 observations, MSE 1.33. Individual labels and dosage colors do not move vertically. This is an exact marginal-shuffle illustration, not a fitted SNP result; the next slide introduces the lab’s approximate conditional permutation.

Combining the forests

Share useful signal among the predictors

A bar totaling OOB R squared 0.4 is divided into temperature 0.3 and rainfall 0.1. Separate bars for other SNPs are combined by averaging each predictor allocation.

Give predictors a share of each SNP’s OOB prediction accuracy, then average over contributing SNPs.

The fitted example

Which gradients carry the most modeled signal?

The six highest-ranked climate predictors by mean allocated out-of-bag R squared in the actual P. guttatus example fit.

BIO18 leads: precipitation of the warmest quarter.

From trees to gradients

From useful splits to a turnover curve

Three aligned environmental axes show useful split locations clustering in the middle, a smoothed density-corrected turnover peak, and a cumulative curve rising most steeply over the same interval.

A cluster of informative splits becomes a steep part of the cumulative curve.

Schematic shapes. Smoothing and density correction account for where samples fall along the gradient.

Timing: 1.5 minutes. Distinguish two uses of importance: OOB permutation importance allocates predictive skill among predictors, while decreases in tree squared-error impurity distribute that allocation along each predictor. With the default density_correction=True, the smoothed split mass is divided by the smoothed observed predictor density for each response, with a floor on the denominator. This corrects uneven environmental coverage in curve construction; it does not remove observation bias, population structure, or spatial dependence. Gaussian smoothing and conditional permutations differ from R gradientForest. Sources: pyGF README Method; general synthesis principle in Ellis et al. 2012. The curves are conceptual rather than numeric transformations of the displayed rug. Coral dashed lines track the same interval through the diagram. Useful split mass is smoothed, corrected for sampling density, integrated, and scaled to allocated importance before averaging across contributing SNPs. Keep the detailed normalization in narration; use the image to explain the transition from a local peak to a cumulative rise.

The fitted example

Read the location and magnitude of turnover

Cumulative turnover curves for the four most important fitted climate variables. Each curve has a nondecreasing shape and an endpoint equal to its mean allocated importance.

Steep segment: faster modeled turnover. Endpoint: total predictor importance. X axes retain each raster’s units.

A new environmental coordinate system

Stretch climate space where turnover is fastest

The same raw climate difference can become a small or large turnover distance.

Illustration. In pyGF, transform() and predict() return these turnover coordinates.

Timing: 1 minute. Each fitted monotone curve transforms its predictor independently. Differences are larger when two raw values straddle a steep portion of that curve. The resulting coordinate differences summarize model-weighted environmental differences associated with genetic variation; they are not a direct estimate of pairwise sequence divergence. Keep the distinction from scikit-learn forest.predict, which would return dosage predictions. Source: pyGF README Python API. The toy raw intervals A–B and C–D are both 0.2 units wide. The first becomes 0.05 turnover units and the second becomes 0.70. Their distances follow interpolation through the displayed piecewise linear curve. This is one environmental coordinate; the real transformation applies each predictor’s own curve and retains one column per predictor.

Project across the landscape

A turnover layer is a transformed climate surface

Regional map of the highest-ranked predictor after transformation through its cumulative turnover curve, with sampled individuals overlaid.

gf.plot_turnover() selects the leading predictor. High values mark cumulative position along that gradient.

Summarizing nineteen turnover coordinates

PCA gives the map and biplot a shared color key

Actual regional map and PCA biplot of GF-transformed environments. Map locations and ordination points share RGB colors derived from the first three principal components.

PC1/2 locate biplot points. PC1/2/3 supply RGB colors. Similar colors indicate similar displayed coordinates.

Reading the biplot

Read the biplot as an environmental compass

An enlarged PCA scatterplot shows nearby similarly colored points and two environmental arrows starting at the origin, with annotations for direction and contribution.

Follow an arrow toward increasing transformed values; compare its contribution to PC1/2.

Schematic PCA scores and covariance arrows. Nearby points share similar scores on the displayed axes.

Aggregate versus one locus

A single SNP can emphasize a different pattern

PCA map and biplot based on turnover curves of the SNP with the highest fitted OOB R squared, using its own PCA and RGB calibration.

Single-SNP view: response=snp. Each view has its own PCA basis and color scale.

Biological interpretation

The same pattern can have different explanations

Two identical white-to-gray-to-black genotype gradients illustrate local adaptation as one hypothesis and shared geography and ancestry as another route to the same observed association.

A climate–genotype association gives us a hypothesis to test.

Conceptual alternatives. Fitness measurements and population-history evidence help distinguish them.

Timing: 1.5 minutes. This is inferential reasoning, not additional evidence collected in the lab. Spatially structured ancestry can align with spatially structured climate without climate causing the genetic pattern. This is especially relevant for only 36 individuals and 19 related predictors. For a spatial prediction claim, refit and evaluate SNP responses in geographic holdouts, with filtering and tuning confined to training information. Genotype prediction error can be evaluated there; the turnover map itself has no directly observed ground-truth color. OOB internal prediction and spatial transfer are different questions. Source for the implementation limits: pyGF README Data and interpretation. Both genotype strips are intentionally identical. The left proposes climate affecting genotype patterns through selection. The right proposes geography relating to climate and ancestry, with ancestry explaining genotype patterns. These are simplified candidate causal structures, not inferred causal graphs. Spatial sampling and correlated environmental variables remain additional alternatives; discuss them using the previous notes.

Gradient Forest

Same climate (predictor) but a different response (SNPs).

Count the alternate copies

Filter for missingness, sufficiently variable sites, LD, and take a subset so it runs faster.

One random forest for each SNP

Evaluating ability to predict genotype of holdout set of samples

Aggregating OOB prediction error across all Trees/SNPs

Some SNPs contribute; others do not

Shuffle climate; see what breaks

Share useful signal among the predictors

Which gradients carry the most modeled signal?

From useful splits to a turnover curve

Read the location and magnitude of turnover

Stretch climate space where turnover is fastest

A turnover layer is a transformed climate surface

PCA gives the map and biplot a shared color key

Read the biplot as an environmental compass

A single SNP can emphasize a different pattern

The same pattern can have different explanations

Further reading