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What are the top 3 methods used to find Autoregressive Parameters in Data Science?
In order to find autoregressive parameters, you will first need to understand what autoregression is. Autoregression is a statistical method used to create a model that describes data as a function of linear regression of lagged values of the dependent variable. In other words, it is a model that uses past values of a dependent variable in order to predict future values of the same dependent variable.
In time series analysis, autoregression is the use of previous values in a time series to predict future values. In other words, it is a form of regression where the dependent variable is forecasted using a linear combination of past values of the independent variable. The parameter values for the autoregression model are estimated using the method of least squares.
The autoregressive parameters are the coefficients in the autoregressive model. These coefficients can be estimated in a number of ways, including ordinary least squares (OLS), maximum likelihood (ML), or least squares with L1 regularization (LASSO). Once estimated, the autoregressive parameters can be used to predict future values of the dependent variable.
To find the autoregressive parameters, you need to use a method known as least squares regression. This method finds the parameters that minimize the sum of the squared residuals. The residual is simply the difference between the predicted value and the actual value. So, in essence, you are finding the parameters that best fit the data.

How to Estimate Autoregressive Parameters?
There are three main ways to estimate autoregressive parameters: ordinary least squares (OLS), maximum likelihood (ML), or least squares with L1 regularization (LASSO).
Ordinary Least Squares: Ordinary least squares is the simplest and most common method for estimating autoregressive parameters. This method estimates the parameters by minimizing the sum of squared errors between actual and predicted values.
Maximum Likelihood: Maximum likelihood is another common method for estimating autoregressive parameters. This method estimates the parameters by maximizing the likelihood function. The likelihood function is a mathematical function that quantifies the probability of observing a given set of data given certain parameter values.
Least Squares with L1 Regularization: Least squares with L1 regularization is another method for estimating autoregressive parameters. This method estimates the parameters by minimizing the sum of squared errors between actual and predicted values while also penalizing models with many parameters. L1 regularization penalizes models by adding an extra term to the error function that is proportional to the sum of absolute values of the estimator coefficients.
Finding Autoregressive Parameters: The Math Behind It
To find the autoregressive parameters using least squares regression, you first need to set up your data in a certain way. You need to have your dependent variable in one column and your independent variables in other columns. For example, let’s say you want to use three years of data to predict next year’s sales (the dependent variable). Your data would look something like this:
| Year | Sales |
|——|——-|
| 2016 | 100 |
| 2017 | 150 |
| 2018 | 200 |
Next, you need to calculate the means for each column. For our sales example, that would look like this:
$$ \bar{Y} = \frac{100+150+200}{3} = 150$$
Now we can calculate each element in what’s called the variance-covariance matrix:
$$ \operatorname {Var} (X)=\sum _{i=1}^{n}\left({x_{i}}-{\bar {x}}\right)^{2} $$
and
$$ \operatorname {Cov} (X,Y)=\sum _{i=1}^{n}\left({x_{i}}-{\bar {x}}\right)\left({y_{i}}-{\bar {y}}\right) $$
For our sales example, that calculation would look like this:
$$ \operatorname {Var} (Y)=\sum _{i=1}^{3}\left({y_{i}}-{\bar {y}}\right)^{2}=(100-150)^{2}+(150-150)^{2}+(200-150)^{2})=2500 $$
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and
$$ \operatorname {Cov} (X,Y)=\sum _{i=1}^{3}\left({x_{i}}-{\bar {x}}\right)\left({y_{i}}-{\bar {y}}\right)=(2016-2017)(100-150)+(2017-2017)(150-150)+(2018-2017)(200-150))=-500 $$
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Now we can finally calculate our autoregressive parameters! We do that by solving this equation:
$$ \hat {\beta }=(X^{\prime }X)^{-1}X^{\prime }Y=\frac {1}{2500}\times 2500\times (-500)=0.20 $$\.20 . That’s it! Our autoregressive parameter is 0\.20 . Once we have that parameter, we can plug it into our autoregressive equation:
$$ Y_{t+1}=0\.20 Y_t+a_1+a_2+a_3footnote{where $a_1$, $a_2$, and $a_3$ are error terms assuming an AR(3)} .$$ And that’s how you solve for autoregressive parameters! Of course, in reality you would be working with much larger datasets, but the underlying principles are still the same. Once you have your autoregressive parameters, you can plug them into the equation and start making predictions!.
Which Method Should You Use?
The estimation method you should use depends on your particular situation and goals. If you are looking for simple and interpretable results, then Ordinary Least Squares may be the best method for you. If you are looking for more accurate predictions, then Maximum Likelihood or Least Squares with L1 Regularization may be better methods for you.
Autoregressive models STEP BY STEP:
1) Download data: The first step is to download some data. This can be done by finding a publicly available dataset or by using your own data if you have any. For this example, we will be using data from the United Nations Comtrade Database.
2) Choose your variables: Once you have your dataset, you will need to choose the variables you want to use in your autoregression model. In our case, we will be using the import and export values of goods between countries as our independent variables.
3) Estimate your model: After choosing your independent variables, you can estimate your autoregression model using the method of least squares. OLS estimation can be done in many statistical software packages such as R or STATA.
4) Interpret your results: Once you have estimated your model, it is important to interpret the results in order to understand what they mean. The coefficients represent the effect that each independent variable has on the dependent variable. In our case, the coefficients represent the effect that imports and exports have on trade balance. A positive coefficient indicates that an increase in the independent variable leads to an increase in the dependent variable while a negative coefficient indicates that an increase in the independent variable leads to a decrease in the dependent variable.
5)Make predictions: Finally, once you have interpreted your results, you can use your autoregression model to make predictions about future values of the dependent variable based on past values of the independent variables.
Conclusion: In this blog post, we have discussed what autoregression is and how to find autoregressive parameters.
Estimating an autoregression model is a relatively simple process that can be done in many statistical software packages such as R or STATA.
In statistics and machine learning, autoregression is a modeling technique used to describe the linear relationship between a dependent variable and one more independent variables. To find the autoregressive parameters, you can use a method known as least squares regression which minimizes the sum of squared residuals. This blog post also explains how to set up your data for calculating least squares regression as well as how to calculate Variance and Covariance before finally calculating your autoregressive parameters. After finding your parameters you can plug them into an autoregressive equation to start making predictions about future events!
We have also discussed three different methods for estimating those parameters: Ordinary Least Squares, Maximum Likelihood, and Least Squares with L1 Regularization. The appropriate estimation method depends on your particular goals and situation.

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Autoregressive Model
Autoregressive generative models can estimate complex continuous data distributions such as trajectory rollouts in an RL environment, image intensities, and audio. Traditional techniques discretize continuous data into various bins and approximate the continuous data distribution using categorical distributions over the bins. This approximation is parameter inefficient as it cannot express abrupt changes in density without using a significant number of additional bins. Adaptive Categorical Discretization (ADACAT) is proposed in this paper as a parameterization of 1-D conditionals that is expressive, parameter efficient, and multimodal. A vector of interval widths and masses is used to parameterize the distribution known as ADACAT. Figure 1 showcases the difference between the traditional uniform categorical discretization approach with the proposed ADACAT.
Each component of the ADACAT distribution has non-overlapping support, making it a specific subfamily of mixtures of uniform distributions. ADACAT generalizes uniformly discretized 1-D categorical distributions. The proposed architecture allows for variable bin widths and more closely approximates the modes of two Gaussians mixture than a uniformly discretized categorical, making it highly expressive than the latter. Additionally, a distribution’s support is discretized using quantile-based discretization, which bins data into groups with similar measured data points. ADACAT uses deep autoregressive frameworks to factorize the joint density into numerous 1-D conditional ADACAT distributions in problems with more than one dimension.
Continue reading | Check out the paper and github link.
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So let's say I have a dataset like MVTec LOCO, which is an anomaly detection dataset specifically for logical anomalies. These are the types of anomalies where some level of logical understanding is required, where traditional anomaly detection methods like Padim and patchcore fail. LVLMs could fill this gap with VQA. Basically a checklist type VQA where the questions are like "Is the red wire connected?" Or "Is the screw aligned correctly?" Or "Are there 2 pushpins in the box?". You get the idea. So I tried a few of the smaller LVLMs with zero and few shot settings but it doesn't work. But then I SFT'd Florence-2 and MoonDream on a similar custom dataset with Yes/No answer format that is fairly balanced between anomaly and normal classes and it gave really good accuracy. Now here's the problem. MVTec LOCO and even real world datasets don't come with a ton of anomaly samples while we can get a bunch of normal samples without a problem because defect happen rarely in the factory. This causes the SFT to fail and the model overfits on the normal cases. Even undersampling doesn't work due to the extremely small amount of anomalous samples. My question is, can we train the model to learn what is normal in an unsupervised method? I have not found any paper that has tried this so far. Any novel ideas are welcome. submitted by /u/SussyAmogusChungus [link] [comments]
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Hello all, I'm wondering if anyone here is on the road to grad school, and if so, how you feel current policy in the United States impacts applications. On one hand, the current administration seems quite adamant about making America "an AI superpower" or whatever, though I think this means bolstering private industry, not universities. They are generally hostile to higher education and ripping away critical funding from schools. Not to mention the hostility towards international students is sure to decrease applicants from abroad. How will this impact (domestic) MS in ML applicants? How will this impact (domestic) PhD applicants? submitted by /u/Zephos65 [link] [comments]
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- arXiv moving from Cornell servers to Google Cloudby /u/sh_tomer (Machine Learning) on April 18, 2025 at 11:31 am
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- Working with distanceby /u/oryx_za (Data Science) on April 18, 2025 at 11:10 am
I'm super curious about the solutions you're using to calculate distances. I can't share too many details, but we have data that includes two addresses and the GPS coordinates between these locations. While the results we've obtained so far are interesting, they only reflect the straight-line distance. Google has an API that allows you to query travel distances by car and even via public transport. However, my understanding is that their terms of service restrict storing the results of these queries and the volume of the calls. Have any of you experts explored other tools or data sources that could fulfill this need? This is for a corporate solution in the UK, so it needs to be compliant with regulations. Edit: thanks, you guys are legends submitted by /u/oryx_za [link] [comments]
- [N] Semantic Memory Layer for LLMs – from long-form GPT interactionby /u/lazylazylazyl (Machine Learning) on April 18, 2025 at 9:57 am
Hi everyone, I’ve spent the past few months interacting with GPT-4 in extended, structured, multi-layered conversations. One limitation became increasingly clear: LLMs are great at maintaining local coherence, but they don’t preserve semantic continuity - the deeper, persistent relevance of ideas across sessions. So a concept started to emerge - the Semantic Memory Layer. The core idea: LLMs could extract semantic nodes - meaning clusters from high-attention passages, weighted by recurrence, emphasis, and user intent. These would form a lightweight conceptual map over time - not a full memory log, but a layer for symbolic relevance and reentry into meaning, not just tokens. This map could live between attention output and decoding - a mechanism for continuity of meaning, rather than short-term prompt recall. This is not a formal proposal or paper — more a structured idea from someone who’s spent a lot of time inside the model’s rhythm. If this connects with ongoing research, I’d be happy to know. Thanks. submitted by /u/lazylazylazyl [link] [comments]
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- [D]Seeking Ideas: How to Build a Highly Accurate OCR for Short Alphanumeric Codes?by /u/ThickDoctor007 (Machine Learning) on April 18, 2025 at 6:54 am
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List of Freely available programming books - What is the single most influential book every Programmers should read
- Bjarne Stroustrup - The C++ Programming Language
- Brian W. Kernighan, Rob Pike - The Practice of Programming
- Donald Knuth - The Art of Computer Programming
- Ellen Ullman - Close to the Machine
- Ellis Horowitz - Fundamentals of Computer Algorithms
- Eric Raymond - The Art of Unix Programming
- Gerald M. Weinberg - The Psychology of Computer Programming
- James Gosling - The Java Programming Language
- Joel Spolsky - The Best Software Writing I
- Keith Curtis - After the Software Wars
- Richard M. Stallman - Free Software, Free Society
- Richard P. Gabriel - Patterns of Software
- Richard P. Gabriel - Innovation Happens Elsewhere
- Code Complete (2nd edition) by Steve McConnell
- The Pragmatic Programmer
- Structure and Interpretation of Computer Programs
- The C Programming Language by Kernighan and Ritchie
- Introduction to Algorithms by Cormen, Leiserson, Rivest & Stein
- Design Patterns by the Gang of Four
- Refactoring: Improving the Design of Existing Code
- The Mythical Man Month
- The Art of Computer Programming by Donald Knuth
- Compilers: Principles, Techniques and Tools by Alfred V. Aho, Ravi Sethi and Jeffrey D. Ullman
- Gödel, Escher, Bach by Douglas Hofstadter
- Clean Code: A Handbook of Agile Software Craftsmanship by Robert C. Martin
- Effective C++
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- CODE by Charles Petzold
- Programming Pearls by Jon Bentley
- Working Effectively with Legacy Code by Michael C. Feathers
- Peopleware by Demarco and Lister
- Coders at Work by Peter Seibel
- Surely You're Joking, Mr. Feynman!
- Effective Java 2nd edition
- Patterns of Enterprise Application Architecture by Martin Fowler
- The Little Schemer
- The Seasoned Schemer
- Why's (Poignant) Guide to Ruby
- The Inmates Are Running The Asylum: Why High Tech Products Drive Us Crazy and How to Restore the Sanity
- The Art of Unix Programming
- Test-Driven Development: By Example by Kent Beck
- Practices of an Agile Developer
- Don't Make Me Think
- Agile Software Development, Principles, Patterns, and Practices by Robert C. Martin
- Domain Driven Designs by Eric Evans
- The Design of Everyday Things by Donald Norman
- Modern C++ Design by Andrei Alexandrescu
- Best Software Writing I by Joel Spolsky
- The Practice of Programming by Kernighan and Pike
- Pragmatic Thinking and Learning: Refactor Your Wetware by Andy Hunt
- Software Estimation: Demystifying the Black Art by Steve McConnel
- The Passionate Programmer (My Job Went To India) by Chad Fowler
- Hackers: Heroes of the Computer Revolution
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Top 1000 Canada Quiz and trivia: CANADA CITIZENSHIP TEST- HISTORY - GEOGRAPHY - GOVERNMENT- CULTURE - PEOPLE - LANGUAGES - TRAVEL - WILDLIFE - HOCKEY - TOURISM - SCENERIES - ARTS - DATA VISUALIZATION

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Health Health, a science-based community to discuss human health
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submitted by /u/Mr_Guavo [link] [comments]
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- The CDC says its tally of US measles cases is likely an undercountby /u/theindependentonline on April 17, 2025 at 7:03 pm
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Today I Learned (TIL) You learn something new every day; what did you learn today? Submit interesting and specific facts about something that you just found out here.
- TIL: that during a dissociative fugue, a person can suddenly travel far from home, assume a new identity, and live for days or even weeks without any memory of their former life.by /u/BedZestyclose3727 on April 18, 2025 at 8:08 am
submitted by /u/BedZestyclose3727 [link] [comments]
- TIL that until the 1970s, Aboriginal children in Australia were systematically taken from their families, known as the Stolen Generationsby /u/ThisIsNotAFarm on April 18, 2025 at 6:48 am
submitted by /u/ThisIsNotAFarm [link] [comments]
- TIL: clouds are 99.9999% air and only 0.0001% water by volume, even though they can weigh thousands of tons.by /u/xk543x on April 18, 2025 at 5:14 am
submitted by /u/xk543x [link] [comments]
- TIL The Cheetah's origins are believed to be in America. Instead of the big cats populating Africa, Asia and (once)Europe, the Cheetah is more closely related to the Puma and the Jaguarundiby /u/Ainsley-Sorsby on April 18, 2025 at 3:42 am
submitted by /u/Ainsley-Sorsby [link] [comments]
- TIL one of the leaders of the NAACP in the early 20th century was Walter White. Who was able to pass as white and protect himself during tense situations in the 20s and 30s.by /u/ILoveTabascoSauce on April 18, 2025 at 3:04 am
submitted by /u/ILoveTabascoSauce [link] [comments]
Reddit Science This community is a place to share and discuss new scientific research. Read about the latest advances in astronomy, biology, medicine, physics, social science, and more. Find and submit new publications and popular science coverage of current research.
- Revolutionary forensic tech gets gunshot residue to glow green | The technology could allow police officers to detect gunshot residue on suspects right at crime scenes, instead of via lab-based tests days later.by /u/chrisdh79 on April 18, 2025 at 10:35 am
submitted by /u/chrisdh79 [link] [comments]
- Most male-female couples who are in satisfying relationships tend to engage in sexual activity close to once per week. 85% of couples reported both high satisfaction and regular sex. Happy sexless couples exist—but they are very rare.by /u/mvea on April 18, 2025 at 10:01 am
submitted by /u/mvea [link] [comments]
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- Research finds evidence of a carbon cycle that operated on ancient Marsby /u/Shiny-Tie-126 on April 18, 2025 at 9:04 am
submitted by /u/Shiny-Tie-126 [link] [comments]
- One-sixth of the planet’s cropland has toxic levels of one or more metalsby /u/nimicdoareu on April 18, 2025 at 5:19 am
submitted by /u/nimicdoareu [link] [comments]
Reddit Sports Sports News and Highlights from the NFL, NBA, NHL, MLB, MLS, and leagues around the world.
- The $10 million club: College basketball's portal recruiting hits unthinkable levels of financial chaosby /u/DontFearTheCreaper on April 18, 2025 at 7:51 am
I'm all for players having the opportunity to at least make SOME cash from their talents. But this shit is out of control, and the game will suffer the longer it takes to install some kind of regulation. submitted by /u/DontFearTheCreaper [link] [comments]
- Yankees' Chisholm blasts umpire on X after ejection, then deletes postby /u/PrincessBananas85 on April 18, 2025 at 4:53 am
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- Canada coach Jesse Marsch ban extended to 2 games by CONCACAF over red card vs USby /u/Oldtimer_2 on April 18, 2025 at 2:22 am
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- Ovechkin adds to record goal total, Crosby dazzles as Penguins top Capitals 5-2 in season finaleby /u/Oldtimer_2 on April 18, 2025 at 2:21 am
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- Everton fan, 62, given lifetime ban from new stadium after filming himself stealing £12.75 worth of chicken stripsby /u/Sandstorm400 on April 17, 2025 at 10:29 pm
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