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| en:iot-reloaded:regression_models [2024/12/02 17:13] – [Piecewise linear models] ktokarz | en:iot-reloaded:regression_models [2024/12/10 21:33] (current) – pczekalski | ||
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| <figure Galton' | <figure Galton' | ||
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| <figure Linear model 1> | <figure Linear model 1> | ||
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| * β0 and β1 y axis crossing and slope coefficients of the linear function correspondingly | * β0 and β1 y axis crossing and slope coefficients of the linear function correspondingly | ||
| - | Unfortunately, | + | Unfortunately, |
| It means that the following equation might describe the model: | It means that the following equation might describe the model: | ||
| <figure Linear model 2> | <figure Linear model 2> | ||
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| <figure Model error> | <figure Model error> | ||
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| <figure Coefficient velues> | <figure Coefficient velues> | ||
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| <figure Galton' | <figure Galton' | ||
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| <figure Coefficient velues> | <figure Coefficient velues> | ||
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| * ei - error of the model' | * ei - error of the model' | ||
| - | Since an error for a given yith might be positive or negative and the model itself minimises the overall error, one might expect that the error is normally | + | Since an error for a given yith might be positive or negative and the model itself minimises the overall error, one might expect that the error is typically |
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| <figure Error_distribution_example> | <figure Error_distribution_example> | ||
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| <figure Error_distribution_example2> | <figure Error_distribution_example2> | ||
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| From this discussion, a few essential notes have to be taken: | From this discussion, a few essential notes have to be taken: | ||
| * Error distributions (around 0) should be treated as carefully as the models themselves; | * Error distributions (around 0) should be treated as carefully as the models themselves; | ||
| - | * In most cases, error distribution is hard to notice even if the errors are illustrated; | + | * In most cases, error distribution is complex |
| * It is essential to look into the distribution to ensure that there are no regularities. | * It is essential to look into the distribution to ensure that there are no regularities. | ||
| If any regularities are noticed, whether a simple variance increase or cyclic nature, they point to something the model does not consider. It might point to a lack of data, i.e., other factors that influence the modelled process, but they are not part of the model, which is therefore exposed through the nature of the error distribution. It also might point to an oversimplified look at the problem, and more complex models should be considered. In any of the mentioned cases, a deeper analysis should be considered. | If any regularities are noticed, whether a simple variance increase or cyclic nature, they point to something the model does not consider. It might point to a lack of data, i.e., other factors that influence the modelled process, but they are not part of the model, which is therefore exposed through the nature of the error distribution. It also might point to an oversimplified look at the problem, and more complex models should be considered. In any of the mentioned cases, a deeper analysis should be considered. | ||
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| <figure Linear model> | <figure Linear model> | ||
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| - | Here, the error is considered to be normally distributed around 0, with its standard deviation sigma and variance sigma squared. Variance provides at least a numerical insight into the error distribution; | + | Here, the error is considered to be normally distributed around 0, with its standard deviation sigma and variance sigma squared. Variance provides at least a numerical insight into the error distribution; |
| <figure Sigma> | <figure Sigma> | ||
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| <figure Variance> | <figure Variance> | ||
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| ===== Multiple linear regression ===== | ===== Multiple linear regression ===== | ||
| - | In many practical problems, the target variable Y might depend on more than one independent variable X, for instance, wine quality, which depends on its level of serenity, amount of sugars, acidity and other factors. In the case of applying a linear regression model, it seems much complicated, but it is still a linear model of the following form: | + | In many practical problems, the target variable Y might depend on more than one independent variable X, for instance, wine quality, which depends on its level of serenity, amount of sugars, acidity and other factors. In the case of applying a linear regression model that doesn' |
| <figure Multiple linear model> | <figure Multiple linear model> | ||
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| <figure Multiple linear model error estimate> | <figure Multiple linear model error estimate> | ||
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| - | Unfortunately, | + | Unfortunately, due to the number of factors (dimensions), the results of multiple linear regression cannot be visualised in the same way as those of a single linear regression. Therefore, numerical analysis and interpretation of the model should be done. In many situations, numerical analysis is complicated and requires a semantic interpretation of the data and model. To do it, visualisations reflecting the relation between the dependent variable and independent variables result in multiple graphs. Otherwise, the quality of the model is hardly assessable or even unassessable. |
| ===== Piecewise linear models ===== | ===== Piecewise linear models ===== | ||
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| <figure Piecewise linear model> | <figure Piecewise linear model> | ||
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| <figure Complex_data_example> | <figure Complex_data_example> | ||
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| <figure Piecewise_linear_model_two> | <figure Piecewise_linear_model_two> | ||
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| <figure Piecewise_linear_model_many> | <figure Piecewise_linear_model_many> | ||
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