He Wang PRO
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Deep neural networks and GW signal recognization
He Wang (王赫)
ITP-CAS, Webniar, Aug 13rd, 2020
Based on: PhD thesis (HTML); 10.1103/PhysRevD.101.104003
Title:
Deep neural networks and GW signal recognization
Abstract:
Deep learning is a neural-inspired pattern recognition technique that is as effective as conventional signal processing. And It has been shown to have considerable potential to identify gravitational-wave (GW) signals. In this talk, I will first review some related works on the detection and characterization of GW signals and some fundamental probabilistic theory of machine learning. I will then present our recent paper (DOI: 10.1103/physrevd.101.104003) about the effect of matched-filtering convolutional neural networks (MFCNN) we proposed on the GW recognition and identifying generalization properties of gravitational waves. At last, I will briefly cover some ongoing works and plans.
Content
- GW astronomy & data analysis
- Challenges and Opportunities
- How machine learning works for GW detection
- A little bit of theory on ML
- Past attempts on stimulated/real LIGO noise
- How MFCNN works for GW detection
- Matched filtering SNR vs Convolution
- Search Results on the O1/O2
- Ongoing works & future plans
Observational Experiment
Theoretical Modeling
Data Analysis
GW astronomy & data analysis
GW astronomy & data analysis
Observational Experiment
Theoretical Modeling
Data Analysis
GW151012
GW170729
GW170809
GW170818
GW170823
GW170121
GW170304
GW170721
(GW151205)
GW Event Detections
O1
O2
O3
GWTC2 (?)
2-OGC (2020)
...
Anomalous non-Gaussian transients, known as glitches
Lack of GW templates
Inadequate matched-filtering method
A threshold is used on SNR value to build our templates bank with a maximum loss of 3% of its SNR.
\text{SNR}=2\left[\int_{0}^{\infty} \frac{|\widetilde{h}(f)|^{2}}{S_{n}(f)} d f\right]^{1 / 2}
Noise power spectral density
Matched filtering Technique:
Optimal detection technique for templates, with Gaussian and stationary detector noise.
credits G. Guidi
GW Detection: Challenges and Opportunities
Real-time / low-latency analysis on raw big data
Anomalous non-Gaussian transients, known as glitches
Lack of GW templates
Inadequate matched-filtering method
The 4-D search parameter space in O1
covered by the template bank
to circular binaries for which the spin of the systems is aligned (or antialigned) with the orbital angular momentum of the binary.
~250,000 template waveforms are used.
The template that best matches GW150914
GW Detection: Challenges and Opportunities
Real-time / low-latency analysis on raw big data
Anomalous non-Gaussian transients, known as glitches
Lack of GW templates
Inadequate matched-filtering method
How many "trash" events?
LIGO L1 and H1 triggers rates during O1
A 'blip' glitch
GW Detection: Challenges and Opportunities
Real-time / low-latency analysis on raw big data
Anomalous non-Gaussian transients, known as glitches
Lack of GW templates
Real-time / low-latency analysis on raw big data
Inadequate matched-filtering method
A new era of multi-messenger astronomy
GW170817: Very long inspiral "chirp" (>100s) firmly detected by the LIGO-Virgo network,
GRB 170817A: 1.74\(\pm\)0.05s later, weak short gamma-ray burst observed by Fermi (also detected by INTEGRAL)
First LIGO-Virgo alert 27 minutes later.
GW Detection: Challenges and Opportunities
Anomalous non-Gaussian transients, known as glitches
Lack of GW templates
Inadequate matched-filtering method
Covering more parameter-space (interpolation)
Automatic generalization to new sources (extrapolation)
Resilience to real non-Gaussian noise (Robustness)
Acceleration of existing pipelines
(Speed, <0.1ms)
-
Phys.Rev.D 97 (2018) 4, 044039
- The first attempt
-
Phys.Rev.Lett 120 (2019) 14, 141103
- Matched-filtering vs CNN
...
-
Phys.Rev.D 100 (2019) 6, 063015
- GW events for BBH
-
Phys.Rev.D 101 (2020) 10, 104003
- All GW events in O1/O2
Why Machine Learning ?
Proof-of-principle studies
Production search studies
Milestones
Real-time / low-latency analysis of the raw big data
More related works, see Survey4GWML (https://iphysresearch.github.io/Survey4GWML/)
GW Detection: Challenges and Opportunities
How machine learning works for GW detection
"A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P, if its performance at tasks in T, as measured by P, improves with experience E."
- What is "learning" by definition? (data-driven algorithm)
Mitchell et al. (1997)
- Tasks:
Whether or not a given noisy data contain a GW signal? (classification problem)
- Measure:
The accuracy of classification: the number of correct predictions made divided by the total number of predictions made. (supervised learning)
- Experience:
Characteristic features of GW waveform. (representations learning)
Sampling rate = 4096 Hz
For a GW sample (1 sec): \(n=4096\)
How machine learning works for GW detection
Dataset containing \(N\) examples sampling from true but unknown data generating distribution :
\mathbf{X}=\left\{\mathbf{x}^{(i)} \mid i=1,2, \cdots, N\right\},
\mathbf{x} \in \mathbb{R}^{n}
with corressponding ground-truth labels:
\mathbf{Y}=\left\{\mathbf{y}^{(i)} \mid i=1,2, \cdots, N\right\},
\mathbf{y} \in \{0, 1\}
Machine learning model is nothing but a map \(f\) from samples to labels:
where \(\mathbb{\Theta}\) is parameters of the model and the outputs are predicted labels:
\hat{\mathbf{Y}}=\left\{\hat{\mathbf{y}}^{(i)} \mid i=1,2, \cdots, N\right\},
0\le \hat{\mathbf{y}} \le 1
p_{\text {data }}(\mathbf{x})
p_{\text {model }}(\hat{\mathbf{y}}|\mathbf{x} ; \boldsymbol{\Theta})
described by , a parametric family of probability distributions over the same space indexed by \(\Theta\).
\mathbf{X} \longrightarrow \hat{\mathbf{Y}}=f(\mathbf{X} ; \Theta)
\mathbf{x} \longmapsto \hat{\mathbf{y}}=f(\mathbf{x} ; \Theta) \in \mathbb{R}
Dataset containing \(N\) examples sampling from true but unknown data generating distribution :
\mathbf{X}=\left\{\mathbf{x}^{(i)} \mid i=1,2, \cdots, N\right\},
\mathbf{x} \in \mathbb{R}^{n}
Sampling rate = 4096 Hz
For a GW sample (1 sec): \(n=4096\)
with coressponding ground-truth labels:
\mathbf{Y}=\left\{\mathbf{y}^{(i)} \mid i=1,2, \cdots, N\right\},
\mathbf{y} \in \{0, 1\}
Machine learning model is nothing but a map \(f\) from samples to labels:
where \(\mathbb{\Theta}\) is parameters of the model and the outputs are predicted labels:
\hat{\mathbf{Y}}=\left\{\hat{\mathbf{y}}^{(i)} \mid i=1,2, \cdots, N\right\},
0\le \hat{\mathbf{y}} \le 1
p_{\text {data }}(\mathbf{x})
p_{\text {model }}(\mathbf{y}|\mathbf{x} ; \boldsymbol{\Theta})
\mathbf{X} \longrightarrow \hat{\mathbf{Y}}=f(\mathbf{X} ; \Theta)
described by , a parametric family of probability distributions over the same space indexed by \(\Theta\).
Objective:
- For each sample,
- Find the best \(\Theta\) that
\hat{\mathbf{y}}^{(i)} \rightarrow \mathbf{y}^{(i)}
p_{\text {model }}(\mathbf{x} ; \boldsymbol{\Theta}) \rightarrow p_{\text {data }}(\mathbf{x})
How machine learning works for GW detection
\mathbf{x} \longmapsto \hat{\mathbf{y}}=f(\mathbf{x} ; \Theta) \in \mathbb{R}
Sampling rate = 4096 Hz
For a GW sample (1 sec): \(n=4096\)
Objective:
- For each sample,
- Find the best \(\Theta\) that
\hat{\mathbf{y}}^{(i)} \rightarrow \mathbf{y}^{(i)}
p_{\text {model }}(\mathbf{x} ; \boldsymbol{\Theta}) \rightarrow p_{\text {data }}(\mathbf{x})
\begin{aligned} \boldsymbol{\theta}_{\mathrm{ML}} &=
\arg \max _{\theta} p_{\text {model }}(\hat{\mathbf{Y}} | \mathbf{X} ; \boldsymbol{\theta}) \\
&=\arg \max _{\theta} \prod_{i=1}^{N} p_{\text {model }}\left(\hat{\mathbf{y}}^{(i)} | \mathbf{x}^{(i)} ; \boldsymbol{\theta}\right) \\
&=\arg \max _{\theta} \sum_{i=1}^{N} \log p_{\text {model }}\left(\hat{\mathbf{y}}^{(i)} | \mathbf{x}^{(i)} ; \boldsymbol{\theta}\right) \\
&=\arg \max _{\boldsymbol{\theta}} \mathbb{E}_{\mathbf{x}, \mathbf{y} \sim \hat{p}_{\text {data }}} \log p_{\text {model }}(\mathbf{y} | \mathbf{x} ; \boldsymbol{\theta})
\end{aligned}
to construct cost function (also called loss func. or error func.)
For classification problem, we always use maximum likelihood estimator for \(\Theta\)
J(\Theta)
FYI: Minimizing the KL divergence corresponds exactly to minimizing the cross-entropy (negative log-likelihood of a Bernoulli/Softmax distribution) between the distributions.
How machine learning works for GW detection
\boldsymbol{J}(\boldsymbol{\theta})=-\mathbb{E}_{\mathbf{x}, \mathbf{y} \sim \hat{p}_{\text {data }}} \log p_{\text {model }}(\mathbf{y} \mid \mathbf{x} ; \theta) \\
\boldsymbol{\theta}_{\mathrm{ML}}=\arg \min _{\theta} \boldsymbol{J}(\theta)
Map / Algorithm
Input
Output
A number
A sequence
Yes or No
\{a_1, a_2, \dots, a_n\}
\in(0 ,1)
\mathbf{x}
\mathbf{y}
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
Our model / network
Extra: ABC of Machine Learning
Past attempts on stimulated noise (1/3)
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- Deeper means better. No more than 3.
Marginal!
Visualization for the high-dimensional feature maps of learned network in layers for bi-class using t-SNE.
Fine-tune Convolutional Neural Network
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
Past attempts on stimulated noise (2/3)
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- Deeper means better. No more than 3.
- A glimpse of model interpretability using visualization.
Extracted features play a decisive role.
\text{Noise} = 0
\text{Noise} > 0
Visualization of the top activation on average at the \(3\)rd layer projected back to time domain using the deconvolutional network approach
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
Marginal!
Past attempts on stimulated noise (2/3)
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- Deeper means better. No more than 3.
- A glimpse of model interpretability using visualization.
Extracted features play a decisive role.
Visualization of the top activation on average at the \(3\)rd layer projected back to time domain using the deconvolutional network approach
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
\text{SNR} = \infty
\text{SNR} = 1
\text{SNR} = 0
Marginal!
Past attempts on stimulated noise (3/3)
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- Deeper means better. No more than 3.
- A glimpse of model interpretability using visualization.
Extracted features play a decisive role.
Occlusion Sensitivity
- Identify what kind of feature is learned.
High sensitivity to the peak features of GW.
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
Marginal!
Past attempts on stimulated noise (3/3)
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- Deeper means better. No more than 3.
Marginal!
- A glimpse of model interpretability using visualization.
Extracted features play a decisive role.
Occlusion Sensitivity
- Identify what kind of feature is learned.
High sensitivity to the peak features of GW.
\mathbf{y} = f(\mathbf{w}\cdot\mathbf{x}+\mathbf{b})
Past attempts on stimulated/real LIGO noise
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
A specific design of the architecture is needed.
[as Timothy D. Gebhard et al. (2019)]
- However, when on real noises from LIGO, this approach does not work that well.
(too sensitive against the background + hard to find GW events)
Past attempts on stimulated/real LIGO noise
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- However, when on real noises from LIGO, this approach does not work that well.
(too sensitive against the background + hard to find GW events)
A specific design of the architecture is needed.
[as Timothy D. Gebhard et al. (2019)]
MFCNN
MFCNN
MFCNN
Our Motivation
- With the closely related concepts between the templates and kernels , we attempt to address a question of:
Matched-filtering (cross-correlation with the templates) can be regarded as a convolutional layer with a set of predefined kernels.
>>Is it matched-filtering ?
>>Wait, It can be matched-filtering!
Classification
Feature extraction
Convolutional neural network (ConvNet or CNN)
- In practice, we use matched filters as an essential component of feature extraction in the first part of the CNN for GW detection.
Matched-filtering in time domain
\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)
(whitening)
where
Time domain
Frequency domain
(normalizing)
(matched-filtering)
\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix}
\bar{d}(t) = d(t) * \bar{S}_n(t) \\
\bar{h}(t) = h(t) * \bar{S}_n(t)
\end{matrix}\right.
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
- The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):
- The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):
\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)
(whitening)
where
Time domain
Frequency domain
(normalizing)
(matched-filtering)
\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix}
\bar{d}(t) = d(t) * \bar{S}_n(t) \\
\bar{h}(t) = h(t) * \bar{S}_n(t)
\end{matrix}\right.
Deep Learning Framework
modulo-N circular convolution
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
Matched-filtering in time domain
- The square of matched-filtering SNR for a given data \(d(t) = n(t)+h(t)\):
\(S_n(|f|)\) is the one-sided average PSD of \(d(t)\)
(whitening)
where
Time domain
Frequency domain
(normalizing)
(matched-filtering)
\langle h|h \rangle = 4\int^\infty_0\frac{\tilde{h}(f)\tilde{h}^*(f)}{S_n(f)}df
\langle d|h \rangle (t) = 4\int^\infty_0\frac{\tilde{d}(f)\tilde{h}^*(f)}{S_n(f)}e^{2\pi ift}df
\rho^2(t)\equiv\frac{1}{\langle h|h \rangle}|\langle d|h \rangle(t)|^2
\langle h|h \rangle \sim [\bar{h}(t) \ast \bar{h}(-t)]|_{t=0}
\langle d|h \rangle (t) \sim \,\bar{d}(t)\ast\bar{h}(-t)
\bar{S_n}(t)=\int^{+\infty}_{-\infty}S_n^{-1/2}(f)e^{2\pi ift}df
\left\{\begin{matrix}
\bar{d}(t) = d(t) * \bar{S}_n(t) \\
\bar{h}(t) = h(t) * \bar{S}_n(t)
\end{matrix}\right.
Deep Learning Framework
modulo-N circular convolution
\int\tilde{x}_1(f) \cdot \tilde{x}_2(f) e^{2\pi ift}df= x_1(t)*x_2(t)
x_1(t)*x_2^*(-t) = x_1(t)\star x_2(t)
\int\tilde{x}_1(f) \cdot \tilde{x}^*_2(f) e^{2\pi ift}df= x_1(t)\star x_2(t)
Matched-filtering in time domain
- In the 1-D convolution (\(*\)), given input data with shape [batch size, channel, length] :
output[n, i, :] = \sum^{channel}_{j=0} input[n,j,:] \ast weight[i,j,:]
FYI: \(N_\ast = \lfloor(N-K+2P)/S\rfloor+1\)
（A schematic illustration for a unit of convolution layer)
Matched-filtering Convolutional Neural Network (MFCNN)
Input
Output
- The structure of MFCNN:
Matched-filtering Convolutional Neural Network (MFCNN)
Input
Output
- The structure of MFCNN:
C_0 = \mathop{\arg\max}_{C}\rho[1,C,N] \,,\\
N_0 = \mathop{\arg\max}_{N} \,\langle d \mid h\rangle[1,C_0,N]
- In the meanwhile, we can obtain the optimal time \(N_0\) (relative to the input) of feature response of matching by recording the location of the maxima value corresponding to the optimal template \(C_0\).
Training Configuration and Search Methodology
- The background noises for training/testing are sampled from a closed set (33*4096s) in the first observation run (O1) in the absence of the segments (4096s) containing the first 3 GW events.
FYI: sampling rate = 4096Hz
- We use SEOBNRE model [Cao et al. (2017)] to generate waveform, we only consider circular, spinless binary black holes.
| template | waveform (train/test) | |
|---|---|---|
| Number | 35 | 1610 |
| Length (s) | 1 | 5 |
| equal mass |
- The total mass and mass ratio of training or test data and templates are shown below. The 11 GW events for both O1 and O2 are also shown.
- Sensitivity estimation (ROC)
True Positive Rate
False Alarm Rate
Training Configuration and Search Methodology
- Every 5 seconds segment as input of our MF-CNN with a step size of 1 second.
- The model can scan the whole range of the input segment and output a probability score.
- In the ideal case, with a GW signal hiding in somewhere, there should be 5 adjacent predictions for it with respect to a threshold.
Training Configuration and Search Methodology
- Every 5 seconds segment as input of our MF-CNN with a step size of 1 second.
- The model can scan the whole range of the input segment and output a probability score.
- In the ideal case, with a GW signal hiding in somewhere, there should be 5 adjacent predictions for it with respect to a threshold.
input
- Recovering three GW events in O1.
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
Search Results on the Real LIGO Recordings
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
- Recovering all GW events in O2, even including GW170817 event.
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
- Recovering all GW events in O2, even including GW170817 event.
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
- Recovering all GW events in O2, even including GW170817 event.
- Our MFCNN can also clearly mark the newly reported GW190412 and GW190814 events in O3a.
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
- Recovering all GW events in O2, even including GW170817 event.
- Our MFCNN can also clearly mark the newly reported GW190412 and GW190814 events in O3a.
-
Statistical significance on O1
- Count a group of adjacent predictions as one "trigger block".
- For pure background (non-Gaussian), monotone trend should be observed.
- In the ideal case, with a GW signal hiding in somewhere, there should be 5 adjacent predictions for it with respect to a threshold.
Number of Adjacent prediction
a bump at 5 adjacent predictions
Search Results on the Real LIGO Recordings
- Recovering three GW events in O1.
- Recovering all GW events in O2, even including GW170817 event.
- Our MFCNN can also clearly mark the newly reported GW190412 and GW190814 events in O3a.
-
Statistical significance on O1
- Count a group of adjacent predictions as one "trigger block".
- For pure background (non-Gaussian), monotone trend should be observed.
- In the ideal case, with a GW signal hiding in somewhere, there should be 5 adjacent predictions for it with respect to a threshold.
-
Glitches identification
- According to GravitySpy Dataset, there are 7368 glitches included in O1 data, 812 of which fall in our trigger set or about 90% of known instrumental glitches is distinguishable.
Conclusions
-
Some benefits from MF-CNN architecture:
-
Simple configuration for GW data generation and almost no data pre-processing.
- It works on a non-stationary background.
-
Easy parallel deployments, multiple detectors can benefit a lot from this design.
- Efficient searching with a fixed window.
-
- The main understanding of the algorithm:
- GW templates can be used as likely features for matching.
- Generalization of both matched-filtering and neural networks.
- Matched-filtering can be rewritten as a convolutional neural layer.
-
Parameter estimation (the current “holy grail” of machine learning for GWs.)
-
Machine learning search for continuous GWs \(\rightarrow\) space-based detector.
-
An improved MFCNN for:
-
higher sensitivity
-
lower FAR (a metric for estimation is urgently needed)
-
and more kinds of GW sources, other than CBC.
-
Ongoing works & future plans
Dr. Chris Messenger (University of Glasgow):
“For me, it seems completely obvious that all data analysis will be ML in 5-10 years".
2008.03312
Deep neural networks and GW signal recognization
By He Wang
Deep neural networks and GW signal recognization
ITP-CAS, Webniar, Aug 13rd, 2020
