In response to these issues, we have developed a calibration pipeline that improves on the strategies presented in Section 2. We introduce a novel Bayesian methodology using conjugate priors for a dynamic application of our algorithm to be run with data collection regardless of system complexity. Also included are model selection methods using machine learning techniques for the optimization of individual noise wave parameters to combat overfitting and underfitting, the results of which converge with that of a least-squares approach when wide priors are adopted.
Our pipeline easily incorporates many more calibrators than the standard four shown in Fig. A schematic of the improved calibration method is shown in Fig. Outline of the Bayesian calibration algorithm. The blue blocks represent data to be taken, red blocks represent calculations, and green blocks represent calculation outputs.
The posterior distribution represents the uncertainty of our parameters after analysis, reflecting the increase in information Nagel As we can see, the posterior distribution is in the same probability distribution family as equation 19 , making our prior a conjugate prior on the likelihood distribution. The resulting numerical computation is many orders of magnitude faster than MCMC methods relying on full numerical sampling and permits an in-place calculation in the same environment as the data acquisition.
This becomes particularly useful for the speed of the algorithm as frequency dependence is introduced in which the computations would not be manageable without conjugate gradients. To verify the performance of our pipeline and highlight features of the algorithm, we evaluate the results of self-consistency checks using empirical models of data based on measurements taken in the laboratory. To make this data as realistic as possible, we used actual measurements of the reflection coefficients of many types of calibrators see Table 1 to generated power spectral densities using equations 6 , 7 , and 8 given a set of realistic model noise wave parameters along with some assumptions about the noise, which are described in Section 3.
The impedances of the calibrators which were measured with a vector network analyser VNA and used in our pipeline are shown on a Smith chart in Fig. Smith chart Argand diagram showing the measured complex impedance of the calibrators used in the Bayesian pipeline across a range of frequencies. We start by demonstrating the importance of correlation between noise wave parameters when determining their values to provide a better calibration solution for the reduction of systematic features in the data such as reflections Section 3.
We then show the increased constraints on these noise wave parameters attributed to the inclusion of more calibrators than the standard number of four Section 3. Following this, we illustrate the effectiveness of model selection for the optimization of individual noise wave parameters to prevent the loss of information resulting from overfitting or underfitting of the data Section 3.
In this section, we show the first major feature of our Bayesian pipeline; the consideration of correlation between noise wave parameters when deriving their values. This is best demonstrated when noise is introduced in an idealized way as to retain a form matching the Gaussian form of our mathematical model. Gaussian noise of one unit variation is then added to the T cal measurements after the calculation to conserve its Gaussian form.
These data are submitted to our algorithm and the resulting posterior distributions for coefficients of the polynomial noise wave parameters are compared to the initial values. Plot showing the joint posteriors of T L and T NS for models using the cold load, the hot load, and both loads concurrently shown as the grey, red, and blue posteriors, respectively. The black cross hairs mark the noise wave parameter values used to generate data submitted to the pipeline.
A zoom-in of the posterior intersection is provided to illustrate the constraint of noise wave parameter values attributed to the correlation between parameters. In Fig. The resulting intersection of posteriors from the individual loads facilitate the derivation of noise wave parameters as the dual-load posterior is found within the region of posterior overlap crossing with the values of the model shown in the inset of Fig.
Retrieval of the noise wave parameter values using correlations between them found in the data demonstrate the relevance of this information which is not taken into account in previous calibration techniques. A nice feature of our pipeline is the ability to include as many calibrators as required to constrain the calibration parameters.
For analysis, six more calibrators are introduced in pairs following the order presented in Table 1. Data for these calibrators are once again generated using fixed terms and Gaussian noise of one unit variation added to T cal as discussed above. Posterior results of our pipeline using data from four, six, and eight calibrators shown in grey, red, and blue, respectively. Cross hairs mark the values of noise wave parameters used to generate the data. We can see that the constraint on noise wave parameter values increases with the number of calibrators used in our pipeline which is encouraging.
Table of calibrators used in the creation of our empirical data models for analysis. Calibrators are added in pairs in the order below when increasing the number of calibration sources used by our algorithm. As shown, the inclusion of more calibrators increases the constraint on the resulting noise wave parameters.
However, we note that after the inclusion of four calibrators, the relative additional constraint decreases with each additional calibrator and thus the use of more than eight calibrators would be unnecessary. The values of noise wave parameters used to generate the data as indicated by the cross hairs in Fig.
The final highlight of our Bayesian pipeline is the use of machine learning techniques to optimize individual noise wave parameters.
This is advantageous as a blanket set of order-seven polynomials applied to all noise wave parameters, such as done in the EDGES experiment, may underfit or overfit individual parameters and misidentify systematics or information about the signal being measured. After multiple iterations, this brings us to the model with the maximal evidence. Data are generated using noise wave parameters as order-2 polynomials. We see that for the model with the highest evidence, that is, the model favoured by the data, the number of polynomial coefficients matches that of the model noise wave parameters.
To demonstrate the robustness of our pipeline, we conducted self-consistency checks using empirically modelled data with a more complicated noise model.
These data were generated using reflection coefficients of eight calibrators and the receiver measured in the laboratory. These reflection coefficients were then smoothed using a cubic smoothing spline Weinert in order to maintain their approximate shape over frequency.
The same second-order noise wave parameters detailed in Section 3. No noise was added to the calibrator input temperatures. This results in a model that does not match the Gaussian form of our mathematical model as in the previous sections and thus does not demonstrate the features of our pipeline as explicitly, but is more representative of data set expected from measurements in the field.
Data for the receiver and the cold load generated using this noise model are shown in Fig. Power spectral densities and reflection coefficients for the receiver and the cold load generated under our realistic noise model. For these higher order tests, we use fgivenx plots which condense noise wave parameter posteriors into samples that can be compared to the model parameter values instead of comparing each individual coefficient Handley Results from samples using data generated with our more realistic noise model shown in black.
The second-order noise wave parameters shown in red are used to generate the data inputted to our pipeline. The polynomial order and values of the noise wave parameters that best suit the data according to our algorithm match that of the empirical model.
By individually adjusting each component of noise arising in our realistic noise model, we may determine what kind of noise our calibration algorithm is most sensitive to, as well as calculate the maximum amount of noise permissible for a specified level of systematic feature reduction. These topics are intended to be explored in a future work. Here we presented the development of a calibration methodology based on the procedure used by EDGES but with key improvements to characterize reflections arising at connections within the receiver.
Our pipeline utilizes the Dicke switching technique and a Bayesian framework in order to individually optimize calibration parameters while identifying correlations between them using a dynamic algorithm to be applied in the same environment as the data acquisition. Future work for the pipeline regards application of real calibrator data, optimization of noise wave parameter coefficients through marginalization techniques and incorporation into an end-to-end simulation based on an entire experimental apparatus to better understand error tolerances.
The flexibility of the algorithm attributed to our novel approach allows its application to any experiment relying on similar forms of calibration such as REACH de Lera Acedo et al. ILVR would like to thank S. Masur for her helpful comments. The data underlying this article will be shared on reasonable request to the corresponding author. Anstey D. Banerjee S. Google Scholar. Google Preview. Barkana R. Bilous A. Bowman J. Cohen A. Dicke R. Ewall-Wice A. Field G. Furlanetto S.
Handley W. The Hunter-VG is reacting! Strange Noise Detected! Abnormal A Button Behavior! The term " Noise Wave " refers to any of a multitude of unique locations in Mega Man Star Force 3 that have spawned as a result of large amounts of Noise. Individual Noise Waves vary in terms of function and parameters; some exhibit the characteristics of a pocket universe , while others act as wormholes. Noise Waves are typically dark, ominous locations, housing a multitude of powerful virii and strange beings known as Noisms , the Noise-based answer to the normal EM Hertz.
Dormant Noise Waves are absolutely invisible to the naked eye, human or digital, and indeed only mildly sensible to EM Beings and technology often there will be a clue, mainly the Hunter-VG screen experiencing slight distortion.
Upon being activated, the Noise Wave entrance blossoms into a dark, pixelating hole through which only EM Beings can enter. When a Noise Wave gate has been activated, it does not revert to total invisibility as before after its use, but a much smaller black shape that is only visible to EM sight, such as exhibited by the Visualizer.
The Hunter-VG terminal comes equipped specifically with the function to indicate and unlock Noise Waves, though they may open accidentally in the presence of other EM beings, such as Hertzes. Once inside the Noise Wave it is typically impossible to escape through the conventional means of Transing Out, which necessitates a search for the exit - those who enter into a Noise Wave are often deposited randomly within the confines of the area, at an unknown distance from the gate.
Noise Waves are also curious in that, by and large, they are accessible only through the Real World, as distinct from natural Cyber Cores accessed from the EM Wave World. As such, it seems that Noise Waves exist on a different possibly subjacent plane of reality than the Real or EM World. There exist four known types of Noise Wave. NOTE: These classifications are unofficial and used solely for distinction purposes.
The first two Noise Wave types are static - according to Omega-Xis who cites a TV Show he once saw , once they have been opened, they will never again shut.
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