What are the top 3 methods used to find Autoregressive Parameters in Data Science?

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.

What are the top 3 methods used to find Autoregressive Parameters in Data Science?
What are the top 3 methods used to find Autoregressive Parameters in Data Science?

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:

$$ \operatorname {Var} (X)=\sum _{i=1}^{n}\left({x_{i}}-{\bar {x}}\right)^{2} $$

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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:

$$ \operatorname {Var} (Y)=\sum _{i=1}^{3}\left({y_{i}}-{\bar {y}}\right)^{2}=(100-150)^{2}+(150-150)^{2}+(200-150)^{2})=2500 $$

and

$$ \operatorname {Cov} (X,Y)=\sum _{i=1}^{3}\left({x_{i}}-{\bar {x}}\right)\left({y_{i}}-{\bar {y}}\right)=(2016-2017)(100-150)+(2017-2017)(150-150)+(2018-2017)(200-150))=-500 $$

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.


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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. 

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    by /u/SussyAmogusChungus (Machine Learning) on April 18, 2025 at 6:20 pm

    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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    by /u/throwaway69xx420 (Data Science) on April 18, 2025 at 5:43 pm

    Hi all, I have found myself numerous times in leadership roles for data science projects. I never feel that I am doing a sufficient job. I find that I either end have up doing a lot of the work on my own and failing to split up task in the data science realm. A lot of these projects, and I hate to say it like this without sounding cocky, I feel that I can do on my own from end to end. Maybe some minimal support from other teams in helping with data flow issues, etc. I'm not a manager by any means, I am individual contributor. For those in this subreddit who are managers, what are some ways you found success in managing data science teams and projects? For those as individual contributors, what are some things that you like to have in a data science manager? submitted by /u/throwaway69xx420 [link] [comments]

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    by /u/Zephos65 (Machine Learning) on April 18, 2025 at 2:49 pm

    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]

  • What’s your 2025 data science coding stack + AI tools workflow?
    by /u/Zuricho (Data Science) on April 18, 2025 at 2:41 pm

    Curious how others are working these days. What’s your current setup? IDE / notebook tools? (VS Code, Cursor, Jupyter, etc.) Are you using AI tools like Cursor, Windsurf, Copilot, Cline, Roo? How do they fit into your workflow? (e.g., prompting style, tasks they’re best at) Any wins, limitations, or tips? submitted by /u/Zuricho [link] [comments]

  • [P] How to handle highly imbalanced biological dataset
    by /u/Ftkd99 (Machine Learning) on April 18, 2025 at 2:40 pm

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  • How do you go about memorizing all the ML algorithms details for interviews?
    by /u/Lamp_Shade_Head (Data Science) on April 18, 2025 at 2:27 pm

    I’ve been preparing for interviews lately, but one area I’m struggling to optimize is the ML depth rounds. Right now, I’m reviewing ISLR and taking notes, but I’m not retaining the material as well as I’d like. Even though I studied this in grad school, it’s been a while since I dove deep into the algorithmic details. Do you have any advice for preparing for ML breadth/depth interviews? Any strategies for reinforcing concepts or alternative resources you’d recommend? submitted by /u/Lamp_Shade_Head [link] [comments]

  • [D] A very nice blog post from Sander Dielman on VAEs and other stuff.
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    Hi guys! Andrej Karpathy recently retweeted a blog post from Sander Dielman that is mostly about VAEs and latent space modeling. Dielman really does a great job of getting the reader on an intellectual journey, while keeping the math and stuff rigorous. Best of both worlds. Here's the link: https://sander.ai/2025/04/15/latents.html I find that it really, really gets interesting from point 4 on. The passage on the KL divergence term not doing much work in terms of curating the latent space is really interesting, I didn't know about that. Also, his explanations on the difficulty of finding a nice reconstruction loss are fascinating. (Why do I sound like an LLM?). He says that the spectral decay of images doesn't align with the human experience that high frequencies are actually very important for the quality of an image. So, L2 and L1 reconstruction losses tend to overweigh low frequency terms, resulting in blurry reconstructed images. Anyway, just 2 cherry-picked examples from a great (and quite long blog post) that has much more into it. submitted by /u/Academic_Sleep1118 [link] [comments]

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    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 interaction
    by /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]

  • Have a lot of experience but not getting any interviews - help
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    Hi, I was here a few weeks back and you helped me to cut down my CV and demo more impact. I have applied to jobs all over and get only rejections. I know the market is hard right now, but I would think that I would at least get invited to have at least initial conversations. This makes me think, there must be something really missing. Could you tell me what you think it could be? Due to AI hype there are a lot of postings with LLMs. I don't have corporate experience there but I plan to do projects to learn & demo it. This week I have lowered my salary requirements by 10k and still get rejections. I have 2 versions - a 2 pager and a 1 pager. Have been applying with the 2 pager mostly until now. Am grateful for your feedback and any help you can give me https://preview.redd.it/e4pubfms4kve1.png?width=1414&format=png&auto=webp&s=853c4ae00db446784cb42ff17048611e5fb03a81 https://preview.redd.it/mzsfifmv4kve1.png?width=1414&format=png&auto=webp&s=ca35aeac336eb834a54b55008efc51936c26658d https://preview.redd.it/l9jz6b6w4kve1.png?width=1414&format=png&auto=webp&s=802f98f4dfdb7cc5d39346c6d1a91cf6b08b95b6 submitted by /u/SonicBoom_81 [link] [comments]

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    Are questions like in 'what if' book, which people rarely bother to ask, way to test whether large language models truly reason, rather than simply remixing patterns and content they see from their training data? submitted by /u/Over_Profession7864 [link] [comments]

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    Hey guys, I’ve been having some issues with Gym Retro. I have installed Gym Retro in PyCharm and have successfully imported Donkey Kong Country into it. From my understanding, Donkey Kong already has a pre-configured environment for Gym Retro to start from, but I don't know how to run the program. Does anyone have a solution? submitted by /u/dbejar19 [link] [comments]

  • [D]Seeking Ideas: How to Build a Highly Accurate OCR for Short Alphanumeric Codes?
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  • Advice before getting data engineer fellowship position
    by /u/Emuthusiast (Data Science) on April 18, 2025 at 3:43 am

    Hey everybody, I need some advice. I have an MsC in Data Science and have really struggled to find jobs. I got an average paying, “data science adjacent but not data science enough” quantitative analyst job in a bank. In fact , I feel like I get dumber every day I’m there and I’m miserable. None of the skills or achievements there are noteworthy : no model building, no big analyses, no data engineering or Gen ai work, just model validation work (helping other people fix their modeling solutions). Long story short, I’m interviewing for a fellowship position to be a data engineer in a nonprofit. It lasts for one year and exposes me to many clients that I will aid. At most I can extend the fellowship for one additional year. It sounds exciting. It pays 10K less, but it’s a step in the right direction. It gets me closer to what I actually studied. The reason I write this post is because I want to know if it will negatively impact my resume or future chances. If I take this job, my resume will look like this : data analyst job (3 years) with a bit of sql and excel, two data science internships (one 3 months and one 8 months) at the university, quantitative analyst (6months), data engineer fellowship (1 year). Will this make companies look at me like a problem and not give me a chance to even interview? Thanks in advance, everybody. submitted by /u/Emuthusiast [link] [comments]

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    Hey, i am working on time series forecasting for the first time . Some information about my data : 30 days data 43200 rows It has two features i.e timestamp and http_requests Time interval is 1 minute I trained LSTM model,followed all the data preprocessing process , but the results are not good and also when i used model for forecasting What would be the reason ? Also how much window size and forecasting step should i take . Any help would be appreciated Thnks submitted by /u/zaynst [link] [comments]

 

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With Google Workspace, Get custom email @yourcompany, Work from anywhere; Easily scale up or down
Google gives you the tools you need to run your business like a pro. Set up custom email, share files securely online, video chat from any device, and more.
Google Workspace provides a platform, a common ground, for all our internal teams and operations to collaboratively support our primary business goal, which is to deliver quality information to our readers quickly.
Get 20% off Google Workspace (Google Meet) Business Plan (AMERICAS): M9HNXHX3WC9H7YE
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Even if you’re small, you want people to see you as a professional business. If you’re still growing, you need the building blocks to get you where you want to be. I’ve learned so much about business through Google Workspace—I can’t imagine working without it.
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