# Help with generalized linear model

• Cexy
In summary, The conversation is about the issue of modelling changes from t to t+1 using some metric d(X(t),X(t+1)). The problem is that x(t) has some dependence on y(t), which needs to be accounted for in the metric. However, x(t) only takes discrete values while y(t) is a positive real number with an unknown distribution. The speaker is seeking suggestions for an appropriate model for the dependence of x(t) on y(t), but acknowledges that it is difficult to model a general stochastic process without knowledge about the underlying physical system.
Cexy
I have a data set {X(t) = (x(t), y(t))}_t=1,...,N and I'm interested in modelling the changes from t to t+1, using some metric d(X(t),X(t+1))

The issue is that x(t) has some dependence on y(t), and I'd like to account for this: if there is a large change in y(t) we expect there to be a corresponding change in x(t), and I'd like the metric to account for this expected change.

The problem is that x(t) takes only discrete values (in fact it is almost always 1 or 2, with a small probability of being any other integer greater than 2) whereas y(t) is a positive real number with an unknown distribution (although of course I can approximate the distribution with a histogram - I have a lot of data available).

What would be an appropriate model for the dependence of x(t) on y(t)?

I've thought about normalizing x(t) to be in the range [0,1] and using a logistic or probit model, but I really have no idea how appropriate this is.

Any ideas?

Cexy said:
...What would be an appropriate model for the dependence of x(t) on y(t)?...

It's difficult to model a general stochastic process by data exploration alone. If anything is known about the underlying physical system, so that appropriate assumptions can be made, it will help in selecting a range of candidate models (possibly including but not limited to GLM).

## 1. What is a generalized linear model (GLM)?

A generalized linear model (GLM) is a statistical model that extends the concept of a linear regression model to accommodate non-normal error distributions and non-linear relationships between the dependent and independent variables.

## 2. When should I use a GLM?

A GLM should be used when the outcome variable is continuous and follows a non-normal distribution, such as a Poisson or binomial distribution. It is also useful when the relationship between the variables is non-linear.

## 3. How do I interpret the results of a GLM?

The results of a GLM can be interpreted in a similar way to a linear regression model. The coefficients represent the change in the dependent variable for a one unit increase in the corresponding independent variable, holding all other variables constant. The significance of the coefficients can be determined through hypothesis testing.

## 4. What are the assumptions of a GLM?

The assumptions of a GLM include linearity, independence of errors, constant variance, and normally distributed errors. However, the GLM can accommodate violations of these assumptions through the use of appropriate error distributions.

## 5. How can I choose the appropriate error distribution for my GLM?

The choice of error distribution for a GLM depends on the type of outcome variable and the underlying data distribution. For example, a Poisson distribution is often used for count data, while a binomial distribution is suitable for binary outcomes. It is important to select the error distribution that best fits the data and satisfies the assumptions of the GLM.

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