# Regression problem with several samples per response (dependent variable)

I have an image dataset in which there is a response $Y$ (dependent variable) for each image. There are about 1000 images, and accordingly 1000 response variable Y per image. My question is how should I define the independent variables since each image contains several pixels but there is only one response value per image. Should I take the average of the color band values (red, green, blue) of all the pixels in one image and consider that the independent variable for that image? This will produce three independent variables and one dependent variable per image. However, if I just take an average over all pixels, I may lose or leave out some information, such as the variance. Is there a machine learning algorithm better than simply taking the average for this type of problem?

If this is an image categorization problem, meaning that the response variable $Y$ is categorical, something like 'dog' or 'cat,' then you should look at a convolutional neural network (CNN). You are correct that merely taking the average will lose quite a lot of information - Think about how many different combinations of the number of pixel values in your RGB channels could lead to the same 'average' value in each channel. Let's say you have $256 \times 256$ pixel images. You thus have $2^{16}$ different pixel values in each of your three channels. Depending on the color spectrum of your $10^3$ samples, that could be quite a few images that are categorically different yet with close average values in each channel. Besides that, small differences in the average value would be interpreted as high similarity in the images, further muddying any inference the model produces.

If this is a true regression problem, meaning that $Y$ is a continuous variable, then you may still have to worry about losing positional information of the pixels by vectorizing the images. I would need more information about the details of this specific problem in order to give any suggestions.

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