Please correct my thinking about Ridge Regression
If ridge regression biases ALL beta coefficients of a regression model towards zero, wouldn't the model massively mispredict the y-variable?
I know my logic must be wrong here, but I'd appreciate if someone could point out the flaw. I am just starting to get into this topic, so would appreciate as simple an answer as possible.
For example: Say you have the coefficients of a linear regression output (and for simplicity, all coefficients are positive, and say the x-variables can only take on positive values). Then, if you biased each beta coefficient towards zero, wouldn't the model consistently underpredict the y-variable?
Would appreciate any intuition on how ridge regression shrinks all the coefficients but still maintains predictive ability. Thank you!
statistics regression mathematical-modeling linear-regression regression-analysis
asked Dec 2 at 17:18
bob
1089
add a comment |
If ridge regression biases ALL beta coefficients of a regression model towards zero, wouldn't the model massively mispredict the y-variable?
I know my logic must be wrong here, but I'd appreciate if someone could point out the flaw. I am just starting to get into this topic, so would appreciate as simple an answer as possible.
For example: Say you have the coefficients of a linear regression output (and for simplicity, all coefficients are positive, and say the x-variables can only take on positive values). Then, if you biased each beta coefficient towards zero, wouldn't the model consistently underpredict the y-variable?
Would appreciate any intuition on how ridge regression shrinks all the coefficients but still maintains predictive ability. Thank you!
statistics regression mathematical-modeling linear-regression regression-analysis
asked Dec 2 at 17:18
bob
1089
add a comment |
If ridge regression biases ALL beta coefficients of a regression model towards zero, wouldn't the model massively mispredict the y-variable?
I know my logic must be wrong here, but I'd appreciate if someone could point out the flaw. I am just starting to get into this topic, so would appreciate as simple an answer as possible.
For example: Say you have the coefficients of a linear regression output (and for simplicity, all coefficients are positive, and say the x-variables can only take on positive values). Then, if you biased each beta coefficient towards zero, wouldn't the model consistently underpredict the y-variable?
Would appreciate any intuition on how ridge regression shrinks all the coefficients but still maintains predictive ability. Thank you!
statistics regression mathematical-modeling linear-regression regression-analysis
asked Dec 2 at 17:18
bob
1089
If ridge regression biases ALL beta coefficients of a regression model towards zero, wouldn't the model massively mispredict the y-variable?
I know my logic must be wrong here, but I'd appreciate if someone could point out the flaw. I am just starting to get into this topic, so would appreciate as simple an answer as possible.
For example: Say you have the coefficients of a linear regression output (and for simplicity, all coefficients are positive, and say the x-variables can only take on positive values). Then, if you biased each beta coefficient towards zero, wouldn't the model consistently underpredict the y-variable?
Would appreciate any intuition on how ridge regression shrinks all the coefficients but still maintains predictive ability. Thank you!
statistics regression mathematical-modeling linear-regression regression-analysis
statistics regression mathematical-modeling linear-regression regression-analysis
asked Dec 2 at 17:18
bob
1089
asked Dec 2 at 17:18
bob
1089
asked Dec 2 at 17:18
bob
1089
asked Dec 2 at 17:18
bob
1089
asked Dec 2 at 17:18
bob
1089
1089
add a comment |
add a comment |
1 Answer
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The objective function in ridge regression has two terms. You are correct that the penalty term $lambda |beta|_2^2$ encourages the coefficients to be small. However, you forget that the main term $|y - X beta|^2$ encourages the choice of coefficients to fit $y$ well. So there is a tension between these two competing goals of fitting $y$ well and keeping the coefficients small.
Regarding "constantly underpredict," you are correct that ridge regression is biased. But this is not the only criteria for evaluating the performance of an estimator (bias-variance tradeoff).
answered Dec 2 at 17:43
angryavian
38.9k23180
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1 Answer
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The objective function in ridge regression has two terms. You are correct that the penalty term $lambda |beta|_2^2$ encourages the coefficients to be small. However, you forget that the main term $|y - X beta|^2$ encourages the choice of coefficients to fit $y$ well. So there is a tension between these two competing goals of fitting $y$ well and keeping the coefficients small.
Regarding "constantly underpredict," you are correct that ridge regression is biased. But this is not the only criteria for evaluating the performance of an estimator (bias-variance tradeoff).
answered Dec 2 at 17:43
angryavian
38.9k23180
add a comment |
The objective function in ridge regression has two terms. You are correct that the penalty term $lambda |beta|_2^2$ encourages the coefficients to be small. However, you forget that the main term $|y - X beta|^2$ encourages the choice of coefficients to fit $y$ well. So there is a tension between these two competing goals of fitting $y$ well and keeping the coefficients small.
Regarding "constantly underpredict," you are correct that ridge regression is biased. But this is not the only criteria for evaluating the performance of an estimator (bias-variance tradeoff).
answered Dec 2 at 17:43
angryavian
38.9k23180
add a comment |
The objective function in ridge regression has two terms. You are correct that the penalty term $lambda |beta|_2^2$ encourages the coefficients to be small. However, you forget that the main term $|y - X beta|^2$ encourages the choice of coefficients to fit $y$ well. So there is a tension between these two competing goals of fitting $y$ well and keeping the coefficients small.
Regarding "constantly underpredict," you are correct that ridge regression is biased. But this is not the only criteria for evaluating the performance of an estimator (bias-variance tradeoff).
answered Dec 2 at 17:43
angryavian
38.9k23180
The objective function in ridge regression has two terms. You are correct that the penalty term $lambda |beta|_2^2$ encourages the coefficients to be small. However, you forget that the main term $|y - X beta|^2$ encourages the choice of coefficients to fit $y$ well. So there is a tension between these two competing goals of fitting $y$ well and keeping the coefficients small.
Regarding "constantly underpredict," you are correct that ridge regression is biased. But this is not the only criteria for evaluating the performance of an estimator (bias-variance tradeoff).
answered Dec 2 at 17:43
angryavian
38.9k23180
answered Dec 2 at 17:43
angryavian
38.9k23180
answered Dec 2 at 17:43
angryavian
38.9k23180
answered Dec 2 at 17:43
angryavian
38.9k23180
38.9k23180
add a comment |
add a comment |
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