Domain inspired machine learning for hypothesis extraction in biological data

Karl Kumbier

UC Berkeley Statistics

Advisor: Prof. Bin Yu

 

In search of mechanistic insights into development and function

  • Genomes contain vast amounts of information (e.g. 6M variants in 20K human genes)

 

  • Complex biological processes are the result of interacting components (e.g. 1e170 potential interactions among human variants)

 

  • Linking genomic processes to biological outcomes requires the exploration of enormous spaces

The PCS framework: evaluating human judgement calls in data science

  • Predictability: Does my model reflect external reality?     ​
  • Computability: Can I tractably build/train my model?           
  • Stability: Are my results consistent with respect to "reasonable" perturbations of the data/model?                         

Outline

  • From genomic to statistical interactions

  • Market baskets and genomics

  • Iterative Random Forests

  • Interaction discovery in Drosophila

From genomic to statistical interactions

Embryonic development in Drosophila

0-1:20 hours

1:20-3:00 hours

3:00-3:40 hours

3:40-5:20 hours

5:20-9:00 hours

9:20-16:00 hours

image: Volker Hartenstein

images: BDGP

 Kr expression

Enhancers regulate spatio-temporal programs of gene expression

Enhancers: segments of the genome that coordinate transcription factor (TF) activity to regulate gene expression.

Experimental evaluation of enhancer elements in Drosophila

Pfeiffer et al. (2008)

even-skipped

expression

wt

transgenic

Experimental evaluation of enhancer elements in Drosophila

Hiromi et al. (1985), Harding et al. (1989), Goto et al. (1989), Pfeiffer et al. (2008)

even-skipped expression

wt

transgenic

High-order interactions at enhancer elements drive embryonic development

Goto et al. (1989), Harding et al. (1989), Small et al. (1992), Isley et al. (2013), Levine et al. (2013)

Identifying regulatory interactions from high-throughput genomic data

Experimentally validated enhancer elements.

Whole-embryo ChIP-chip/ChIP-seq measurements of transcription factor (TF) DNA binding

From genomic to statistical interactions

r(\mathbf{x})=
\prod_{j\in\text{A}} 1(x_j > t_j)
\cdot
\prod_{j\in\text{R}} 1(x_j \le t_j)

activators

repressors

\mathbf{x}=(x_1, \dots, x_p)

Segment of the genome

DNA binding for p transcription factors (TFs)

Order-s interaction: s = #activators + #repressors

r(\mathbf{x}) = 1
x_{bcd}
x_{cad}
\dots

RuleFit: rule-based interaction discovery (Friedman and Popescu, 2008)

  1. Identify a collection of marginally important features
  2. Search for predictive order-2 rules among marginally important features
\dots

Computational costs grow as

O(p^s)

Misses interactions with weak  marginal effects

image: Lee and Haber (2014)

Thresholding rules define expression domains

Chopra and Levine (2009)

Dl +

Dl -

Wolpert (1968), Jaeger and Reinitz (2006), Chopra and Levine (2009), Zizen et al. (2009), Knowles and Biggin (2013), Levine (2013), Staller al. (2015), ...

Jaeger and Reinitz (2009)

From genomic to statistical interactions

\mathbb{P}_n(y = 1 | S)
\mathbb{P}_n(S|y = 1)

(1) How precisely does an interaction predict class-1 observations?

(2) How prevalent is an interaction among class-1 observations?

S = (x_{j_1} > t_{j_1}) \,\&\, \dots \,\&\, (x_{j_s} \le t_{j_s})

Interactions:

Responses:

y \in \{0, 1\}
r(\mathbf{x})=

?

Market baskets and genomics 

Interactions in market baskets

What combinations of items do customers purchase together?

Interactions in market baskets

What combinations of items do customers purchase together?

What combinations of items do different types of customers purchase together?

Interactions in market baskets

Z_1
Z_2
Z_3
Z_4
\mathcal{I}_1
\mathcal{I}_2
\mathcal{I}_3
\mathcal{I}_4

Feature-index sets

Random intersection trees (RIT)

Shah and Meinshausen (2014)

Leverage sparsity in market baskets to search for frequently co-occurring items in a computationally efficient manner

  1. Randomly sample feature index sets from class-C observations:
  2.  Intersect sampled feature index sets in a tree like fashion up to depth D
  3. Return all feature combinations that "survive" intersection procedure up to depth D
\{\mathcal{I}_i \subseteq\{1, \dots, p\}: Z_i = C\}

Random intersection trees (RIT)

Shah and Meinshausen (2014)

Randomly sampled

class-C observation

"survived" interaction

Random intersection trees (RIT)

Shah and Meinshausen (2014)

Random intersection trees (RIT)

Shah and Meinshausen (2014)

Genomic response

Genomic features

Translating the market basket problem into genomics

\dots

Genomic response

Genomic features

Translating the market basket problem into genomics

Challenges:

  1. Genomic features are typically measured in concentrations/counts
  2. Binding does not imply regulation (Li et al. 2008)
\dots

iterative Random Forests (iRF)

&

signed iterative Random Forests (siRF)

Joint work with Sumanta Basu, James B. Brown, Susan Celniker, and Bin Yu

 iterative Random Forest to identify high-order interactions in genomic data

  1. Iteratively re-weighted Random Forests stabilize decision path                       
  2. Generalized random intersection trees search for high-order interactions         
  3. Stability bagging evaluates interactions                           

iterative Random Forests (iRF) build on PCS to identify genomic interactions in developing Drosophila embryos 

Open source R implementation: https://cran.r-project.org/web/packages/iRF/

Iteratively re-weighted random forests

Classification and regression trees (CART)

Breiman et al. (1984)

 

For current node:

  1. Select splitting feature and threshold
  2. Partition data 
  3. Repeat until stopping criteria
x_2>t_2
x_3>t_3
x_4>t_4

Classification and regression trees (CART)

Breiman et al. (1984)

 

For current node:

  1. Select splitting feature and threshold
  2. Partition data 
  3. Repeat until stopping criteria
x_2>t_2
x_3>t_3
x_4>t_4
(x_2 > t_2) \,\&\,(x_3 > t_3) \,\&\,(x_4 > t_4)

Random Forests

Breiman (2001)

Random forests modify CART to improve predictive accuracy:

  1. CART trees are trained on bootstrap samples of the data
  2. CART criterion evaluated on subset of features sampled uniformly at random

The CART criterion: Gini impurity

(\pi, N)
(\pi_l, N_l)
(\pi_r, N_r)

Proportion positive responses

Number of observations

Gini impurity:

I_G(\pi) = \pi (1-\pi)

Decrease in Gini impurity:

I_G(\pi)-\frac{N_l}{N}I_G(\pi_l) - \frac{N_r}{N}I_G(\pi_r)

Mean decrease in impurity:

On average, how much does splitting on a variable decrease the Gini impurity?

Feature-weighted Random Forests

Amaratunga et al. (2008)

Random forests: 

At each node of the decision tree, uniformly sample a subset of features

Feature-weighted random forests: 

At each node of the decision tree, sample a subset of features with probability proportional to 

w\in\mathbb{R}^p_+
w_1
w_2
w_3
w_4
w_5
w_1
w_2
w_3
w_4
w_5

Feature weights

Iterative re-weighting stabilizes random forest decision paths

Gini importance

Iteration 1

Iteration K

w_1
w_2
w_3
w_4
w_5
w_1
w_2
w_3
w_4
w_5

Feature weights

Iterative re-weighting helps recover high-order interactions

Generalized random intersection trees

Encoding decision paths to extract active features

Active

Inactive

Continuous measurements

Binary features

Encoding decision paths to extract enriched and depleted features

Enriched

Depleted

Generalized random intersection trees search for high-order interactions

\cap
\cap
\cap
\cap
\{1,4\}

1. Iteratively re-weighted random forests

3. RIT on random forest decision paths

2. Decision path feature transformation

\mathbf{x}_1 \rightarrow \mathcal{I}_1 =
\mathbf{x}_n \rightarrow \mathcal{I}_n =

.

.

.

\emptyset

Encoding decision paths to extract prevalent decision rules

Continuous measurements

Binary feature encoding

Decision rules

Encoding decision paths to extract prevalent decision rules

\mathbf{x}_i
S_{i_1}
S_{i_T}

. . .

Generalized random intersection trees search for high-order interactions

Prevalent interactions

Binary feature encoding

RIT

. . .

Runtime comparison between iRF and RuleFit

Evaluating interactions

Importance measures for high-order interactions

  • Prevalence: how frequently is an interaction observed among positive responses?

 

 

 

 

  • Precision: how accurately does an interaction predict positive responses?

 

 

P(S|y=1)=\frac{1}{T} \sum_{t=1}^T \frac{\sum_{i=1}^n I(S\subseteq S_{i_t}) I(y_i = 1)}{\sum_{i=1}^n I(y_i = 1)}
P(y=1|S)=\frac{1}{T} \sum_{t=1}^T \frac{\sum_{i=1}^n I(S\subseteq S_{i_t}) I(y_i = 1)}{\sum_{i=1}^n I(S\subseteq S_{i_t})}

Null metrics describe importance measures under simple structure

 

  • Class precision enrichment (CPE): is an interaction more prevalent among a particular class?

 

  • Feature selection dependence (FSD): are features selected in a dependent manner?

 

  • Mean increase in precision (MIP): do additional features improve prediction accuracy?

Screening interactions based on simple structure (CPE)

No

Yes

Is an interaction enriched in one class relative to the other?

Prevalence measures the stability of an interaction across an RF

Prevalence: 

P(S|y=1)=\frac{1}{T} \sum_{t=1}^T \frac{\sum_{i=1}^n I(S\subseteq \mathcal{I}_{i_t}) \cdot I(y_i = 1)}{\sum_{i=1}^n I(y_i = 1)}
P(\{1, 3, 4\}|y = 1) =3/5
P(\{1, 4\}|y = 1) =4/5

Examples:

Precision measures the predictive accuracy of an interaction across an RF

Precision:

P(y=1|S)=\frac{1}{T} \sum_{t=1}^T \frac{\sum_{i=1}^n I(S\subseteq \mathcal{I}_{i_t}) \cdot I(y_i = 1)}{\sum_{i=1}^n I(S\subseteq \mathcal{I}_{i_t})}
P(y=1|\{1, 4\}) = 4/6
P(y=1|\{1, 3, 4\}) = 3/4

Examples:

Screening interactions based on simple structure (FSD)

Do features co-occur in a dependent manner?

No

Yes

Screening interactions based on simple structure (MIP)

Do additional features allow provide predictive power?

No

Yes

Bagging evaluates stability of interactions across entire iRF workflow relative to resampling

\cap
\cap
\cap
\cap
\cap
\cap
\cap
\cap
\{1,4\}
\{1,3\}
P(S|y=1)
P(S|y=1)
\dots

1. Iteratively re-weighted RF stabilize decision paths

2. gRIT searches for high-order interactions along decision paths

3. Importance metrics evaluate interactions in fitted RF

Outer layer bootstrap samples

\emptyset
\{1,4\}

Case studies in Drosophila

Predicting enhancer activity throughout embryonic development

Enhancers: Pfeiffer et al. 2008, Fisher et al. 2012, Kvon et al. 2014

ChIP: MacArthur et al. 2009, Li et al. 2008, modENCODE/modERN consortia

x_1
x_2
\dots
y

Predicting enhancer activity throughout embryonic development

Early stage (not shown): 24 TFs; Basu, K., Brown, and Yu (2018)

 

All stages (shown): 307 TFs; K., Basu, Brown, Celniker, and Yu (2019)

iterative Random Forests recover well-known pairwise interactions

Signed interactions define active enhancers with high precision

RF-prediction

0

1

Interaction-based predictions

Signed interactions define active enhancers with high precision

RF-prediction

0

1

Interaction-based predictions

High false

negative rate

Missing

data (stage)

TFs involved in important interactions are measured at "high confidence" stages

Activated late

Activated early

Novel order-3 interactions exhibit AND-like behavior (D, Hth, Kni)

Novel order-3 interactions exhibit AND-like behavior (Hth, Kni, Kr)

Putative interactions define targets for follow-up experiments

High confidence enhancer elements

Proposed mechanism: Hth+ & Kni+ & Kr+

Summary

  • iRF and siRF identify well known interactions in Drosophila and posit new, high-order interactions surrounding important regulatory factors in the early embryo. 
  • By decoupling interaction order from the computational cost of discovery, iRF and siRF allow us to investigate mechanisms in genome biology and beyond. 

Ackowledgements

S. Basu

J. Brown

B. Yu

S. Celniker

E. Frise

Yu Group

Thank You!

siRF

By kkumbier

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