# 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).

## Finding Autoregressive Parameters: The Math Behind ItTo 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

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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:

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

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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.

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• [D] f1/fbeta vs average precision
by /u/ActiveBummer (Machine Learning) on June 18, 2024 at 1:30 pm

Both f1/fbeta and average precision take into account recall and precision. These are great options to combine recall and precision into a single metric for optimization of model performance. Are there any situations where one would be preferred to the other? And why? I could think of average precision being easier to explain to non technical stakeholders. Keen to hear your thoughts! submitted by /u/ActiveBummer [link] [comments]

• [D] Optimising metric and satisficing metric
by /u/ActiveBummer (Machine Learning) on June 18, 2024 at 1:24 pm

It makes sense to set an acceptance criteria for satisficing metric. After all, the model has to meet the acceptance criteria for deployment. For example, suppose biz requirement is precision has to be at least 80%. Precision is the satisficing metric. Models with precision less than 80% will not be deployed. Optimizing metric means the model has to be as good as possible for this metric. For example, average precision. (Tuning classification threshold to improve precision trades off recall, and vice versa. We want to reduce such trade off over all possible thresholds and hence look for a metric that maximises PRAUC i.e. average precision.) For optimizing metric, does it make sense to have a threshold? What purpose does this threshold bring when it doesn't influence model acceptance or rejection? Keen to hear your thoughts! submitted by /u/ActiveBummer [link] [comments]

• [D] Transfer Learning given differently calibrated measurement equipment
by /u/SmokinCaterpill4r (Machine Learning) on June 18, 2024 at 12:09 pm

We have the following problem, let us assume we are looking at some machine measurements X of a production sample S and some expensive quality test of the sample Y in some plant A. We want to predict the outcome Y from X for other samples for which we cannot afford doing Y. Together X and Y form a perfect tabular dataset, perfect for most classical machine learning tasks with some caveats. First, the dataset a rather small (~500 to 5000 samples), as measuring Y is expensive, and measurements X can deteriorate or change over time or even show sudden shifts due to machine calibration. So concept drift is the norm rather the exception. Now we also have data from plant B, who measure X with a different device, but also do the expensive test Y. At least we can rely on the outcome Y, as this test is standardized across plants. So if we measured Y for sample S in plant A, this would give the same results if measured in plant B. However, the devices measuring X aren't standardized and up for manually calibration. So there might be shifts in mean values or slight shifts in the general scales compared across the plants A and B. What are good strategies to compensate for the small training set by pooling data from plant A and B? What methods exist to counteract mean or scale shifts in X between plant A and B? Note we CANNOT send around a calibration sample S and measure X in both plants to get an idea of the shifts and differences in measuring X. Anyone knowing some good papers or approaches for this kind of transfer learning? Thank You very much! submitted by /u/SmokinCaterpill4r [link] [comments]

by /u/petrichorinforest (Machine Learning) on June 18, 2024 at 12:01 pm

I’m a data science undergrad dreaming of making a mark in research. I’m drawn to reinforcement learning (RL), computer vision (CV), and optimization. Yet, I’m unsure which is more suited for Quant job/healthcare. My university specializes in deep learning for CV and evolutionary algorithms but not in healthcare applications. Should I focus on RL or CV for a healthcare research direction, or is the evolutionary algorithm pathway promising? Any advice would be greatly appreciated. submitted by /u/petrichorinforest [link] [comments]

• [News] Athens NLP Summer School, September 19-25 2024
by /u/yannisassael (Machine Learning) on June 18, 2024 at 10:30 am

• [Project] How to create effective multimodal retrieval system for Multimodal RAG?
by /u/badtemperedpeanut (Machine Learning) on June 18, 2024 at 8:31 am

Lets say you need to retrieve Images and text based on user query, I think you can take 2 approaches. What would be a better approach? Is there an even better approach? Appproach 1: Convert everything into embeddings, search based on the embeddings. Approach 2: Get a textual description from images, convert that text into embeddings and search the text based embeddings. In case of Approach 2 there is an added benefit of having an option to combine keyword based search. submitted by /u/badtemperedpeanut [link] [comments]

• [R] Scaling Synthetic Logical Reasoning Datasets with Context-Sensitive Declarative Grammars
by /u/Jean-Porte (Machine Learning) on June 18, 2024 at 8:16 am

• [D] Evolutionary Strategy vs. Backpropagation
by /u/kiockete (Machine Learning) on June 18, 2024 at 8:00 am

I was looking into Evolutionary Strategy for training NNs and I'm getting pretty interestnig results. Here is the notebook you can play with: Link to the Colab Notebook Number of epochs Final Accuracy Seconds per epoch Backpropagation 10 97% 9 Evolutionary Strategy 10 90% 9 I wonder how far it can be pushed, but getting 90% of accuracy for something that does not use gradient information at all and completes the training within the same amount of time on GPU as backpropagation is quite interesting. The ES algorithm used is very simple: Initialize all weights with zeros Create new generation population of size N - draw every weight from normal distribution where mean is the current weight and standard deviation is the learning rate. Calculate loss for every individual in population in parallel - works very well on GPUs Pick top-k best performing individuals for mating. To get next weight tensor for new generation take a mean of top-k best performing individuals. Go to step 2. Do you know of any cool research that explores Evolutionary Strategies for training neural networks? submitted by /u/kiockete [link] [comments]

• [D] Recommendation for open source RAG frameworks
by /u/AccomplishedBar5572 (Machine Learning) on June 18, 2024 at 6:32 am

I was building a personal project with basic RAG functionality, and I started looking into using some open source framework to improve the retrieval pipeline. What are good open source frameworks that people are using? R2R, RagFlow, Canopy, ...? Does anyone have experience with any of them and what was your experience with them? submitted by /u/AccomplishedBar5572 [link] [comments]

• End-to-end project feedback
by /u/-S-I-D- (Data Science) on June 18, 2024 at 6:09 am

Hi, I am planning to create an end-to-end ML project to showcase my skillsets end to end. I have finished the process of getting raw data, cleaned it, EDA and then created an ML model. Now I would like to go forward with the next step which is to deploy it locally and then on the cloud, here are the steps I was thinking of doing and would appreciate any feedback or suggestions if my approach is wrong: Save model using “Pickle” Create an app.py file for Flask to create an API endpoint Test if the API works locally using Postman. Create HTML and Javascript files for interaction with the Flask API and display the prediction in the front-end. I've also seen ppl porting the data that I used to created the model into a SQL database. Any reason why this should be done? Is this part of CI/CD? After the above steps work properly, should I then start with deploying it on the cloud? I plan to deploy it on Azure cloud since that is commonly used in my country. Also I want to try out using Model Deployment Tools since that is what is commonly used by companies since they allow for easier scaling, monitoring etc. so I want to learn and showcase this part as well. Should I work on this part after I finish deploying it on the cloud? submitted by /u/-S-I-D- [link] [comments]

• Multiple data in a same timestamp [D]
by /u/dumbestindumb (Machine Learning) on June 18, 2024 at 6:02 am

Hello everyone, I'm here to clarify my doubts. For a project, I have connected five routers to create a mesh network. From that, I have collected some data between each link of the routers. The problem is that since I collected it all at the same time, I have multiple data points with the same timestamp. As I'm new to time series analysis, I'm not sure how to handle such situations. Please suggest a good solution if you have encountered similar situations. submitted by /u/dumbestindumb [link] [comments]

• [R] How would I approach this NLP problem?
by /u/SnooMaps8602 (Machine Learning) on June 18, 2024 at 4:52 am

How would I approach this NLP problem? So i’m thinking of using data that is conversation between multiple people labeled with big five personality traits (represented by numbers) to identify someone’s personality. I’m using the BERT model and a FriendsPersona dataset, implementing a paper, but apparently BERT can only work well on simple monologue text. The accuracy also isn’t that great. Does anyone know any models or papers that I can implement to approach this problem, perhaps with a bit of transfer learning? submitted by /u/SnooMaps8602 [link] [comments]

• [R] New survey and review paper for video diffusion models!
by /u/MolassesWeak2646 (Machine Learning) on June 18, 2024 at 4:22 am

Title: Video Diffusion Models: A Survey Authors: Andrew Melnik, Michal Ljubljanac, Cong Lu, Qi Yan, Weiming Ren, Helge Ritter. Paper: https://arxiv.org/abs/2405.03150 Abstract: Diffusion generative models have recently become a robust technique for producing and modifying coherent, high-quality video. This survey offers a systematic overview of critical elements of diffusion models for video generation, covering applications, architectural choices, and the modeling of temporal dynamics. Recent advancements in the field are summarized and grouped into development trends. The survey concludes with an overview of remaining challenges and an outlook on the future of the field. https://preview.redd.it/1845dt7zca7d1.png?width=2496&format=png&auto=webp&s=17f52b44e9c3f1f06fe66e784512adae3d1c20de submitted by /u/MolassesWeak2646 [link] [comments]

• Holdout testing vs. Cross validation
by /u/WhiteRaven_M (Data Science) on June 18, 2024 at 3:53 am

When i build models, i typically make a train, val, and test set. I'll train models off of the train set, tune them based on validation set, and the final test accuracy is what I use to estimate its performance. I understand the argument for cross validation, it averages out variation in different samples so the estimation of model performance is closer to reality. But on larger datasets, 1) isnt this not really an issue because your samples are larger therefore central limit theorem yada yada 2) more computationally expensive? Am i missing something? submitted by /u/WhiteRaven_M [link] [comments]

• [R] DiTTo-TTS: Efficient and Scalable Zero-Shot Text-to-Speech with Diffusion Transformer
by /u/keonlee9420 (Machine Learning) on June 18, 2024 at 3:51 am

• Do data scientists use t-tests to prove that their model performances are better than other models?
by /u/limedove (Data Science) on June 18, 2024 at 3:11 am

I understand that it can be stated that the result of 80% accuracy is better than a 79% accuracy. But maybe that is just for one sample set of data coming from a population. So maybe after some more accuracy calculations and seeing how much the accuracy results vary, you can say that the 80% vs the 79% don't differ really significantly statistically. submitted by /u/limedove [link] [comments]

• [D]Did anyone receive a desk rejection warning in the ARR June (EMNLP) cycle?
by /u/ImpossibleAd568 (Machine Learning) on June 18, 2024 at 2:46 am

I submitted to ARR June (EMNLP) and received the following message. None of the authors have an ACL Anchology profile, and some of the authors have only had 1 paper accepted this year (not yet published). I think, we are all ineligible to be reviewers(volunteer) (since we don't have at least 3 papers). Does anyone have any idea why we received this message? submitted by /u/ImpossibleAd568 [link] [comments]

• Chat GPT / Copilot productivity hacks you use in your day to day job?
by /u/LikkyBumBum (Data Science) on June 18, 2024 at 12:43 am

What GPT productivity hacks do you use to make your work day easier? For example, I use it to create massive long annoying case / if else statements. I just paste the data into chat gpt and say: turn this into a case statement for me. I also use it to find errors in my code if I can't figure it out in less than 10 seconds. Super basic and boring. Yes I know it can help me code in general. I feel like I'm not using it to its full potential. Any cool hacks you'd like to share? submitted by /u/LikkyBumBum [link] [comments]

• [D] Batch-norm behavior with bounded activation function
by /u/zhbug (Machine Learning) on June 17, 2024 at 11:46 pm

Hi, This prior post discusses Batch-norm with Relu ordering. I have a similar question but pertaining to sigmoid or tanh kind of activation functions. Should batch-norm be used before or after? Is there a difference? We add the BN transform immediately before the nonlinearity, by normalizing x = Wu+ b. We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity, the shape of its distribution is likely to change during training, and constraining its first and second moments would not eliminate the covariate shift. In contrast, Wu + b is more likely to have a symmetric, non-sparse distribution, that is “more Gaussian” (Hyvarinen & Oja, 2000); normalizing it is likely to produce activations with a stable distribution. ^ Quote from the original BN paper. In the BN paper they seem to have the batch-norm before the sigmoid and evaluate it on MNIST dataset and show improved results. It may be more appropriate after the activation function if for s-shaped functions like the hyperbolic tangent and logistic function. ^ However, this BN article claims that putting BN after sigmoid might be better. Doesn't give any justification... Has anyone thought about this or played around with this? In my current use case, I see that BN after is improving some of my results. I do not understand any why. I wrote this code to visualize some differences. I understand why the output looks as it does, but I do not understand how it carries over to the learning process. https://preview.redd.it/2oq4mbrex77d1.png?width=1416&format=png&auto=webp&s=29c43a90b483a8da55d154d62e447bcd96640cb8 https://preview.redd.it/xn5rpnffx77d1.png?width=1419&format=png&auto=webp&s=7df3afb01ca7866673482c29c83dc2f71a857d58 import numpy as np import matplotlib.pyplot as plt import torch import torch.nn.functional as F # Generate a random uniform distribution D np.random.seed(0) D = np.random.uniform(-3, 3, 1000) # Function to apply sigmoid and layer normalization def sigmoid_then_layernorm(data): sigmoid_data = torch.sigmoid(torch.tensor(data, dtype=torch.float32)) layernorm = torch.nn.LayerNorm(sigmoid_data.shape) normalized_data = layernorm(sigmoid_data) return normalized_data.detach().numpy() # Function to apply layer normalization and then sigmoid def layernorm_then_sigmoid(data): data_tensor = torch.tensor(data, dtype=torch.float32) layernorm = torch.nn.LayerNorm(data_tensor.shape) normalized_data = layernorm(data_tensor) sigmoid_data = torch.sigmoid(normalized_data) return sigmoid_data.detach().numpy() # Apply both methods sigmoid_then_layernorm_data = sigmoid_then_layernorm(D) layernorm_then_sigmoid_data = layernorm_then_sigmoid(D) # Plot the distributions plt.figure(figsize=(14, 6)) plt.subplot(1, 2, 1) plt.hist(sigmoid_then_layernorm_data, bins=50, alpha=0.7, label='Sigmoid then LayerNorm') plt.title('Sigmoid then LayerNorm') plt.xlabel('Value') plt.ylabel('Frequency') plt.legend() plt.subplot(1, 2, 2) plt.hist(layernorm_then_sigmoid_data, bins=50, alpha=0.7, label='LayerNorm then Sigmoid') plt.title('LayerNorm then Sigmoid') plt.xlabel('Value') plt.ylabel('Frequency') plt.legend() plt.tight_layout() plt.show() # Generate a normal distribution D D = np.random.normal(0, 1, 1000) # Apply both methods sigmoid_then_layernorm_data = sigmoid_then_layernorm(D) layernorm_then_sigmoid_data = layernorm_then_sigmoid(D) # Plot the distributions plt.figure(figsize=(14, 6)) plt.subplot(1, 2, 1) plt.hist(sigmoid_then_layernorm_data, bins=50, alpha=0.7, label='Sigmoid then LayerNorm') plt.title('Sigmoid then LayerNorm') plt.xlabel('Value') plt.ylabel('Frequency') plt.legend() plt.subplot(1, 2, 2) plt.hist(layernorm_then_sigmoid_data, bins=50, alpha=0.7, label='LayerNorm then Sigmoid') plt.title('LayerNorm then Sigmoid') plt.xlabel('Value') plt.ylabel('Frequency') plt.legend() plt.tight_layout() plt.show() submitted by /u/zhbug [link] [comments]

• Putting models into production
by /u/HumerousMoniker (Data Science) on June 17, 2024 at 11:28 pm

I'm a lone operator at my company and don't have anywhere to turn to learn best practices, so need some help. The company I work for has heavy rotating equipment (think power generation) and I've been developing anomaly detection models (both point wise and time series), but am now looking at deploying them. What are current best practices? what tools would help me out? The way I'm planning on doing it, is to have some kind of model registry, and pickle my models to retain the state, then do batch testing on new data, and store results in a database. It seems pretty simple to run it on a VM and database in snowflake, but it feels like I'm just using what I know, rather than best practices. Does anyone have any advice? submitted by /u/HumerousMoniker [link] [comments]

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