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#### Motivation
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- "Understand" the relationship between physical (brain) and mental properties
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- Question 1: are the two related at all?
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- Question 2: how are they related?
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- Note: many efforts dedicated to Q1, fewer to Q2.
- We will formally address Q1, and address Q2
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#### One often desires to test for independence
| \\(X\\) | \\(Y\\) |
| :---: | :---: |
| clouds | grass wetness |
| brain connectivity | creativity |
| brain shape | health |
| CLARITY | condition |
| gene expression | cancer |
--
| anything | anything else |
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#### Formal Definition of Independence Testing
$$(X\_i,Y\_i) \sim F\_{XY} = F\_{X|Y} F\_Y, \quad i \in \{1,\ldots,n\}$$
$$H\_0: F\_{XY} = F\_X F\_Y $$
$$H\_A: F\_{XY} \neq F\_X F\_Y $$
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## Outline of Talk
- intuition
- simulations
- theory
- real data
- discussion
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# intuition
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#### Intuitive Desiderata of Testing Procedure
1. Performant under *any* joint distribution
- low- and high-dimensional
- Euclidean and structured data (eg, sequences, images, networks, shapes)
- linear and nonlinear relationships
6. Reveals the "geometry" of dependence
5. Is computational efficiency
Provides a tractable algorithm that addresses the two motivating questions:
- Question 1: are the two related at all?
- Question 2: how are they related?
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### correlation coefficient
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
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### **mantel** correlation coefficient
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
$$d\_{XY}^2 = \frac{(\sum\_{i,\color{red}{j}=1}^n ( x\_i - \color{red}{x\_j}) (y\_i - \color{red}{y\_j}))^2}{\sum\_{i,\color{red}{j}=1}^n (x\_i - \color{red}{x\_j})^2 \sum\_{i,\color{red}{j}=1}^n (y\_i- \color{red}{y\_j})^2} $$
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### **generalized** correlation coefficient
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
$$d\_{XY}^2 = \frac{(\sum\_{i,\color{red}{j}=1}^n ( x\_i - \color{red}{x\_j}) (y\_i - \color{red}{y\_j}))^2}{\sum\_{i,\color{red}{j}=1}^n (x\_i - \color{red}{x\_j})^2 \sum\_{i,\color{red}{j}=1}^n (y\_i- \color{red}{y\_j})^2} $$
$$c\_{XY}^2 = \frac{(\sum\_{i,j=1}^n \color{red}{\sigma\_x}(x\_i,x\_j) \color{red}{\sigma\_y}(y\_i,y\_j))^2}{\sum\_{i,j=1}^n \color{red}{\sigma\_x}(x\_i,x\_j)^2 \sum\_{i,j=1}^n \color{red}{\sigma\_y}(y\_i,y\_j)^2} $$
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### local distance correlation
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dcorr(X,Y)=0.15, p-val < 0.001
MGC(X,Y)=0.15, p-val < 0.001
--
--
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dcorr(X,Y)=0.01, p-val 0.3
MGC(X,Y)= .r[0.13], p-val < .r[0.001]
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### multiscale distance correlation
- compute local dcorr **at all scales**
- find scale with **max** smoothed test statistic
- permutation test to determine p-value
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### Multiscale Generalized Correlation (MGC)
--
--
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#### Intuitive Desiderata of Testing Procedure
1. Performant under *any* joint distribution
- low- and high-dimensional
- Euclidean and structured data (eg, sequences, images, networks, shapes)
- linear and nonlinear relationships
6. Reveals the "geometry" of dependence
5. Is computational efficiency
Provides a tractable algorithm that addresses the two motivating questions:
- Question 1: are the two related at all?
- Question 2: how are they related?
---
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# simulations
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### Definitions
- **power** is the probability of rejecting the null when the alternative is true
- $N_\beta(t)$ := the # of samples required to achieve power $\beta$ using test statistic $t$
### Empirical Desiderata
1. statistical efficiency: $$N\_\beta(t') / N\_\beta( \text{MGC}) \geq 1 \text{ for all }
n, \, d, \, F\_{XY}, \text{ and } t' $$
2. quantification of relationship
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### 20 Different Functions (1D version)
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### MGC Outperforms Benchmarks
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### MGC Outperforms Benchmarks
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#### MGC Reveals Geometry of Dependence
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### Empirical Desiderata
1. $N\_\beta(t') / N\_\beta( \text{MGC}) \geq 1 \text{ for all }
n, \, d, \, F\_{XY}, \text{ and } t' $
2. quantification of relationship
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# theory
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### Theoretical Desiderata
| name | principle |
| --- | --- |
| boundedness | $0 \leq t,T \leq 1 $|
| symmetric | $T(X,Y) = T(Y,X)$ |
| 1-linear. | $ T= 1 \Leftrightarrow y = A x + b$ |
| 0-indep. | $T=0 \Leftrightarrow H(X \lvert Y) = H(X)$ |
| ortho. invar. | $T(X,Y) = T(a_1 + b_1 C_1 X, a_2 + b_2 C_2 Y)$ |
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| univ. consist. | $\beta\_n(T) \to 1, \quad \forall \, F\_{XY}$ |
| dominance | $\beta\_n(T) \geq \beta\_n(T'), \, \forall n,\, T' \in \mathcal{T}, \forall F\_{XY}$ |
| convergence | $\beta_n(t) \to \beta_n(T)$ as $n \to \infty$ |
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#### MGC is a Reasonable Dependence Statistic
Thm: Oracle MGC has the following properties:
- 0 ≥ MGC ≥ 1
- MGC = 0 only under independence
- MGC is symmetric
- MGC = 1 only under linear relationship
- MGC is invariant to rotation, translation, and scale of X and/or Y
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### MGC has power 1 for all FXY
Lemma: \\( \beta_n (T^*) \to 1 \\) as \\( n \to \infty \\) whenever \\( E [X] < \infty. \\)
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### Linear: Local < Global
Lemma: If x is linearly dependent on y, then it always holds that
$$ \beta_n(T^*) = \beta_n(T) $$
for any n.
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### (Certain) Nonlinear: Local > Global
Lemma: There exists f and n such that
$$\beta_n( T^{k,l} ) > \beta_n(T). $$
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### MGC > Dcorr
Thm: Oracle MGC statistically dominates dcorr, that is,
$$\beta_n( T^* ) \geq \beta_n(T). $$
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### Sample MGC Converges
Thm: For any FXY with finite 1st moment,
$\beta_n(t^*) \to \beta_n(T^\ast)$ as $n \to \infty$
if and only if $\sigma_X$ and $\sigma_Y$ are of strong negative type.
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### Theoretical Desiderata
| name | principle |
| --- | --- |
| boundedness | ✅|
| symmetric | ✅|
| linear. | ✅ |
| 0-indep. | ✅ |
| ortho. invar. | L$_p$ |
| univ. consist. | ✅ |
| dominance | ✅ |
| convergence | ✅ |
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# real data
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### Real Data Desiderata
1. when we believe dependence, MGC obtains a small p-value
2. MGC provides insight into the geometry of real dependence
3. when there is no dependence, MGC correcly controls FPR
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### MGC Discovers Relationships between Brain & Mental Properties
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### MGC Reveals Geometry of Real Data
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### Overall Summary
- Oracle MGC theoretically dominates, even in finite samples
- MGC empirically nearly dominates on extensive simulations
- Visual quantitative characterization of arbitrary relationships
- MGC reveals geometry of dependence in real data
- MGC mitigate "post selection inference" problems
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### Some Extensions
- Foundational theory for MGC [[1]](https://arxiv.org/abs/1710.09768)
- MGC for independence between graph topology & attributes [[2]](https://arxiv.org/abs/1703.10136)
- MGC for signal subgraph detection [[3]](https://arxiv.org/abs/1801.07683)
- MGC for clustering (ish) [[4]](https://arxiv.org/abs/1710.09859)
- MGC for K-sample testing
- MGC for feature selection
- MGC for batch effect detection
- MGC for effect size characterization
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### Next Steps
- Make R package from open source code ([github link](https://github.com/neurodata/mgc))
- Scale up for larger n
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## Questions?
e: [jovo@jhu.edu](mailto:jovo@jhu.edu)
w: [neurodata.io/tools#MGC/](http://neurodata.io/tools#MGC)
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### Computational Desiderata
1. fast
2. open source
| Method | Complexity |
| :---: | :---: |
| Dcorr | n2 |
| HHG | n2 log n |
| MGC | n2 log n / T |
- MATLAB and R code at [http://neurodata.io/tools#MGC](http://neurodata.io/tools#MGC)