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?

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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.

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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).

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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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$$ \operatorname {Cov} (X,Y)=\sum _{i=1}^{n}\left({x_{i}}-{\bar {x}}\right)\left({y_{i}}-{\bar {y}}\right) $$

For our sales example, that calculation would look like this:

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


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

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:

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$$ Y_{t+1}=0\.20 Y_t+a_1+a_2+a_3footnote{where $a_1$, $a_2$, and $a_3$ are error terms assuming an AR(3)} .$$ 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!

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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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Continue reading | Check out the paper and github link.

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  • [D] How do people keep up with ML news that is not NLP related?
    by /u/shaner92 (Machine Learning) on January 29, 2023 at 9:48 am

    Lately, NLP is taking up most of the public space, much of AI news is focused on LLM after Chat-GPT took the spotlight. How do non-NLP people keep up with news? I recently saw a post on reddit where tree models are still being improved. There are other topics too, like the recent trend in Model Explainability which feels to have slowed down. I'd guess this all gets into the more categorical questions which I am wrapping up with 'How do YOU get your ML news'? How does information gathering differ between those in Applied ML and AI researchers (or even further, between those in Business Analytics and those in more 'AI' fields) What sort of interesting things are out there in the world of ML now? (model or non-model related) In looking for Use Cases, does this partially come down to your field? (Finance reads finance news, pharma reads pharma news) ​ Many of the AI/ML Newsletters which I subscribed to when I was less experienced seemed to be full of variety, but as they are all converging to NLP recently maybe it is time to cleanse the subscriptions, or find some new resources. submitted by /u/shaner92 [link] [comments]

  • What test to use to test differences in proportions of total?
    by /u/zsa23761 (Data Science) on January 29, 2023 at 6:46 am

    Let’s say I’m trying to increase the proportion of total female customers on my site. If I test 2 landing pages and my target metric is proportion of total purchases that are women, what test should I use? Let’s say we have the following sample data: Control: 400 male customers, 600 female customers Treatment: 350 male customers, 600 female customers What test can I use to test whether the increase in female proportion is statistically significant? submitted by /u/zsa23761 [link] [comments]

  • A new way to accelerate your data science workflow
    by /u/CORNMONSTER_2022 (Data Science) on January 29, 2023 at 5:54 am

    When working with large datasets or executing complex operations using pandas, you may experience Out of Memory (OOM) errors or long waiting periods for the results. Xorbits can be an ideal solution for these issues. Xorbits is a scalable Python data science framework that aims to scale the Python data science stack while keeping the API compatibility. You can get an out-of-box performance gain by changing `import pandas as pd` to `import xorbits.pandas as pd`. Compared with other pandas-like solutions, Xorbits offers stronger performance, simpler deployment, and better API compatibility. With TPC-H benchmarks at scale factor 100, Xorbits is 7x faster than Dask. For the benchmark results compared to other pandas-like systems, please visit: https://xorbits.io/benchmark https://preview.redd.it/uyavcs2irqea1.png?width=2353&format=png&auto=webp&s=455d690ca78bfe69a3e05abc0dd0d09ca24dded6 If you are interested in learning more about Xorbits, please visit our project's Github for more information: https://github.com/xprobe-inc/xorbits submitted by /u/CORNMONSTER_2022 [link] [comments]

  • Has anyone gotten a job with a data science certificate?
    by /u/madlove17 (Data Science) on January 29, 2023 at 5:24 am

    I'm in the U.S. and I've enrolled in the WIOWA (Workforce Innovation and Opportunity Act) program since I'm a dislocated worker. I'm interested in getting a cert through UCLA's Extension course program. If you've gotten certs what was your experience? submitted by /u/madlove17 [link] [comments]

  • Advice on Learning Pathway(s) and where personal interests intersect with DS
    by /u/SilentWitness13 (Data Science) on January 29, 2023 at 5:05 am

    Hi folks, Currently working through the IBM DS and Google Analytics certificate. I am a complete rookie in DS. I am interested in where linguistics and behavioral economics intersect with data science in terms of the types of jobs/research fields to target as it seems like 'Data Scientist' is somewhat of a catch all term for HR departments. In terms of a following a learning pathway, employers (HR?) seem to all look for a degree, however online postgrad and BsS computer science course seem overpriced and, outside of data science specific programs, not as good as combining bootcamps, certificates and other online learning routes. Is this the case? In terms of building a portfolio of work, or a learning journal that can be linked to CV, does anyone have any recommendations? Many thanks for any help! submitted by /u/SilentWitness13 [link] [comments]


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