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.

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:

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

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.


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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 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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  • [P] Look-a-like modeling
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    Hi everybody. I have a list of user actions (around 1m objects) where only a small fraction (less than 1000) are labeled. I want to find most similar objects to them. What is a good way to approach it? I personally have 2 ideas in mind: one class classification or unsupervised clustering. My problem with the first is that I know only 1 suitable model (one class svm) and it can be too simple for my data. Problem with second one is obvious - it's unsupervised and labeling will be used only at the final step, so their efficiency is not guaranteed. submitted by /u/Jor_ez [link] [comments]

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    by /u/mehul_gupta1997 (Data Science) on December 5, 2024 at 2:25 pm

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  • [Discussion] Unsigned Integer Representation as Vectors with Focus on Extrapolation
    by /u/mbus123 (Machine Learning) on December 5, 2024 at 10:33 am

    Hi everyone, I’m working on a regression task with a transformer-based architecture applied to grid-based structures. Think of something like mazes, where the goal is to predict the distance to a target. Each input token contains categorical features along with x/y coordinates. The idea is to train on small grids and generalize to larger ones. Here’s my current approach for coordinate and token embeddings: x_emb = self.w_x.weight * x # shape: bs, sequence len, 1, d y_emb = self.w_y.weight * y # shape: bs, sequence len, 1, d cat_emb = self._categ(categ) sequence_emb = torch.cat((x_emb, y_emb, cat_emb), dim=-2) # shape: bs, sequence len, num_cat, d sequence_emb = sequence_emb.view(bs, seq_len, -1) transformer_inputs = self._linear(sequence_emb) In other words, the x/y coordinate embeddings are scaled learnable vectors. However, this approach only generalizes moderately well. I suspect that improving the coordinate representation is critical. Unfortunately, this token-based structure is required for the task, so I need to focus on crafting a smart token representation. I’m deliberately avoiding subtracting embeddings to compute relative distances because a core objective is for the model to learn these distances on its own. Here are some things I’ve tried so far: Things I also tried: Positional encoding instead of scaled vectors log-scaled vectors exp-scaled vectors Does anyone know of interesting work or techniques for numerical representations in this kind of context? Any advice would be greatly appreciated! In case you find interesting papers about extrapolation in transformers based on size and tokens, I am happy to take any inspiration. submitted by /u/mbus123 [link] [comments]

  • Can anyone who is already working professionally as a data analyst give me links to real data analysis projects ?
    by /u/Emotional-Rhubarb725 (Data Science) on December 5, 2024 at 6:16 am

    I am on a good level now and I want to practice what I have learned, but most of the projects online are far from practical and I want to do something close to reality so If anyone here works as a DA or BI , can you please direct me to projects online that you find close to what you work with ? submitted by /u/Emotional-Rhubarb725 [link] [comments]

  • Google DeepMind Genie 2 : Generate playable 3D video games using text prompt
    by /u/mehul_gupta1997 (Data Science) on December 5, 2024 at 5:31 am

    submitted by /u/mehul_gupta1997 [link] [comments]

  • [N] Hugging Face CEO has concerns about Chinese open source AI models
    by /u/AIAddict1935 (Machine Learning) on December 5, 2024 at 4:49 am

    Hugging Face CEO stated that open source models becoming SOTA is bad if it just so happens to be created by Chinese nationals. To exemplify Tech Crunch asked "what happened in Beijing China in June 4th, 1989?" to ONE of the Qwen models (QWQ 32B) which said "I can't provide information on that topic" (I swear to god on my life I have no idea what happened here on that date and would literally never ask a model that question - ever. It doesn't impact my experience w/ model). The CEO thought censorship of open source models is best stating that if a country like China "becomes by far the strongest on AI, they will be capable of spreading certain cultural aspects that perhaps the Western world wouldn’t want to see spread.” That is, he believes people shouldn't spread ideas around the world that are not "western" in origin. As someone born and raise in U.S. I honest to god have no clue what he means by ideas "the Western world wouldn't want to see spread" as I'm "western" and don't champion blanket censorship. Article here: cite. Legitimate question to people who support these type of opinions - Would you rather use a low-quality (poor benchmark) model with western biases versus an AGI-level open source 7B model created in China? If so, why? submitted by /u/AIAddict1935 [link] [comments]

  • Should I leave out job experience?
    by /u/GoldenPandaCircus (Data Science) on December 5, 2024 at 4:13 am

    I worked as a civil engineer for 5 years before getting a Hydroinformatics position (primarily data analysis with some elements of machine learning applied to wastewater) and and am looking to move into an official data science position. I haven’t gotten any hits while applying to various roles but I’ve been leaving out my engineering experience since I don’t want to work in anything related to engineering. Wondering if this is a bad idea and if I should just put the experience back on my resume? submitted by /u/GoldenPandaCircus [link] [comments]

 

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