Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times . Then, is said to be stationary if, for all , for all , and for all ,
Since does not affect , is not a function of time.
In all cases, the estimation target is a function of the independent variables called the regression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by a probability distribution.
The performance of regression analysis methods in practice depends on the form of the data generating process, and how it relates to the regression approach being used. Since the true form of the data-generating process is generally not known, regression analysis often depends to some extent on making assumptions about this process. These assumptions are sometimes testable if a sufficient quantity of data is available. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However, in many applications, especially with small effects or questions of causality based on observational data, regression methods can give misleading results.
Adaptive subtraction: Standard adaptive subtraction methods use the well-known minimum energy criterion, stating that the total energy after optimal multiple attenuation should be minimal.
where represents the non-stationary convolution with the multiple model obtained with SRMP (i.e., Chapter ) and are the input data. These filters are estimated in a least-squares sense for one shot gather at a time. Note that in practice, a regularization term is usually added in equation () to enforce smoothness between filters. This strategy is similar to the one used in Chapter . The residual vector contains the estimated primaries.
In this section, the multiple model computed in the preceding section is subtracted from the data with two techniques. The model is obtained after shot interpolation with the sparseness constraint. The first technique is a pattern-based method introduced in Chapter that separates primaries from multiples according to their multivariate spectra. These spectra are approximated with prediction-error filters. The second technique adaptively subtract the multiple model from the data by estimating non-stationary matching filters (see Chapter ).
Shaping regularization: http://www.reproducibility.org/RSF/book/jsg/shape/paper_html/
Least squares sense:
Least Squares Fitting
A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets (“the residuals”) of the points from the curve. The sum of the squares of the offsets is used instead of the offset absolute values because this allows the residuals to be treated as a continuous differentiable quantity. However, because squares of the offsets are used, outlying points can have a disproportionate effect on the fit, a property which may or may not be desirable depending on the problem at hand.
In signal processing, a matched filter (originally known as a North filter) is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal to noise ratio (SNR) in the presence of additive stochastic noise. Matched filters are commonly used in radar, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal. Pulse compression is an example of matched filtering. It is so called because impulse response is matched to input pulse signals. Two-dimensional matched filters are commonly used in image processing, e.g., to improve SNR for X-ray. Matched filtering is a demodulation technique with LTI filters to maximize SNR.
Derivation of the matched filter impulse response
The matched filter is the linear filter, , that maximizes the output signal-to-noise ratio.
Though we most often express filters as the impulse response of convolution systems, as above (see LTI system theory), it is easiest to think of the matched filter in the context of the inner product, which we will see shortly.
We can derive the linear filter that maximizes output signal-to-noise ratio by invoking a geometric argument. The intuition behind the matched filter relies on correlating the received signal (a vector) with a filter (another vector) that is parallel with the signal, maximizing the inner product. This enhances the signal. When we consider the additive stochastic noise, we have the additional challenge of minimizing the output due to noise by choosing a filter that is orthogonal to the noise.
Let us formally define the problem. We seek a filter, , such that we maximize the output signal-to-noise ratio, where the output is the inner product of the filter and the observed signal .
Our observed signal consists of the desirable signal and additive noise :
Let us define the covariance matrix of the noise, reminding ourselves that this matrix has Hermitian symmetry, a property that will become useful in the derivation:
We now define the signal-to-noise ratio, which is our objective function, to be the ratio of the power of the output due to the desired signal to the power of the output due to the noise:
We rewrite the above:
We wish to maximize this quantity by choosing . Expanding the denominator of our objective function, we have
Now, our becomes
We will rewrite this expression with some matrix manipulation. The reason for this seemingly counterproductive measure will become evident shortly. Exploiting the Hermitian symmetry of the covariance matrix , we can write
We would like to find an upper bound on this expression. To do so, we first recognize a form of the Cauchy-Schwarz inequality:
which is to say that the square of the inner product of two vectors can only be as large as the product of the individual inner products of the vectors. This concept returns to the intuition behind the matched filter: this upper bound is achieved when the two vectors and are parallel. We resume our derivation by expressing the upper bound on our in light of the geometric inequality above:
Our valiant matrix manipulation has now paid off. We see that the expression for our upper bound can be greatly simplified:
We can achieve this upper bound if we choose,
where is an arbitrary real number. To verify this, we plug into our expression for the output :
Thus, our optimal matched filter is
We often choose to normalize the expected value of the power of the filter output due to the noise to unity. That is, we constrain
This constraint implies a value of , for which we can solve:
giving us our normalized filter,
If we care to write the impulse response of the filter for the convolution system, it is simply the complex conjugate time reversal of .
Though we have derived the matched filter in discrete time, we can extend the concept to continuous-time systems if we replace with the continuous-time autocorrelation function of the noise, assuming a continuous signal , continuous noise , and a continuous filter .
I will give you but three words, “FORESIGHT”,”DILIGENCE”,”STEADFASTNESS”, If you ask me to summarize what I took away from Saranya Murthy’s talk at UT AWM (American Women in Mathematics)’s weekly talks by successful and influential women in STEM in academia or industry. Saranya Murthy is an International Product Support Associate Engineer at Dell. She was a formal employee at Workbrain(now Infor), a software used by employees, providing web-based workforce management solutions for large enterprises. She graduated from University of Waterloo in Honors program in computer science in the faculty of mathematics. Later on after her graduation and working for years, she accomplished a part-time MBA program at York University in Canada. Miss Murthy must be originally from India, as shown from her skin and that her parents who are traditionally dressed in Sari(an Indian female garment) and Lungi are also in the audience while she’s giving us the talk. She maintained a graceful demeanour from beginning to end. She was very clear in logic- she gave us useful suggestions(I will demonstrate them later in this blog) about how to build personal development and why those ways are efficient and helpful; she was very precise about diction too, which can indirectly reflect her work ethics and lifestyles. Later after the presentation she told me she loves reading and holds book clubs with her friends (recently they’re reading “Lean in” written by Sheryl Sandberg- Chief Operation Officer at FB. Accidentally I am reading this book too. I’m currently on the first chapter about the gap between women’s ambition and leadership).
I want to specifically note down the advice given by Saranya Murthy during her talk and thus to benefit my readers.They’re as follows:
1. 360º feedback –http://www.reachcc.com/360reach -Click on 360º Reach Basic- it’s free!
2.MBI test(Apply the rest results to interviews and occasion of the same nature)
3.”Please understand me” by Dr. David Keirsey
4.”And Now Discover your Strengths” by Marcus Buckingham and Donald O.Clifton.
HOW TO BE A “CARE-FOR-YOUR-FUTUER” UNDERGRADUATE STUDENT
1.When deciding to registering for a class, ask the professor also for yourself “How can I use this in the real world?”;
2.Connect with like-minded professionals through information interviews and relevant social events;
3.Thoughtfully craft a 30 second elevator pitch which you may use later to pique people’s interest and let them quickly experience your charisma;
4.Develop a professional social media persona:
-Linkedin (Employers will usually check for Linkedin for candidates’ information and they particularly pay attention to your coworkers’/mentors’/professors’ recommendations and comments on it)
-Twitter (Thought leader) -WordPress blogging (why I’m blogging now:=)
Before the conclusion of my blog, I also want to point out that Miss Murthy is not only a role model in terms of work ethics and academic excellence, but more importantly she is a role model for me in terms of how respectful and obedient she is towards her parents. I approached Saranya for answers and guidance to my specific questions right after she concluded her talk, when her elderly parents are waiting for her besides. She friendly and politely addressed to me that we’d better find a place to sit down and talk so that on the one hand we can talk in details and on the other hand my parents can sit down and wait for me. These details of her bearing unintentionally touched my heart. I recalled how I yelled at my parents when I’m unhappy and how I unconsciously ignore them when I’m with friends…… I’d love to end my first blog at WordPress with Miss Murthy’s self-summaries on Linkedin which can constantly remind me of what precious personality my role model has and what I’m supposed to act to acquire that kind of personality and charisma in near future.
“• 9 years of experience in Quality Assurance (QA) for enterprise software
• Demonstrated leadership skills – motivated individual who can take charge and drive change
• Strong understanding of Software Development Life Cycle and relevant QA concepts
• Excellent technical and analytical skills – experienced at troubleshooting software issues and tracing defect triggers
• Strong experience in defining and improving software test processes
• Proven Project Management skills – responsible for overseeing QA team to deliver thorough and timely test efforts for 3-month release cycle
• Enthusiastic team player with effective interpersonal and leadership skills
• Experience working in Cross-Functional Teams – representing QA and liaising with Developers, Product Managers and Technical Writers
• Client-oriented approach: well-honed ability to test software from the “user’s point of view”
• Excellent written and verbal communication skills”