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How to turn your dog’s nap time into a connected linear model

When you’re architecture a machine learning model you’re faced with the bias-variance tradeoff, where you have to find the antithesis amid having a model that:

  1. Is very alive and captures the real patterns in the data.
  2. Generates predictions that are not too far off from the actual values,

A model that is very alive has a low bias, but it can also be too complex. While a model that generates predictions that aren’t too far off from the true value has low variance.


When the model is too circuitous and tries to encode more patterns from the training data than it’s absolutely necessary, it will start acrimonious up on noise. Since it’s already going beyond the real patterns of the data, the model is overfitting the data.

Overfitting usually happens when you have a small training set, too small to advertise all the nuance and patterns in the data. With only a small amount of data to learn from, the model ends up hyper-focusing only on the patterns it can see in the training set. As a consequence, the model is not generalizable, acceptation it won’t be very good at admiration the targets of data it has never seen before.

An overfit model is also a non-generalizable model, because it’s fine tuned to adumbrate the targets of the data in the training set.

This is where regularization comes in! You can use regularization techniques to ascendancy overfitting and create a model with high predictive accurateness while befitting it as simple as possible.

In practice, regularization tunes the values of the coefficients. Some coefficients can end up with such a negligible addition to the model or even be equal to zero and you can confidently ignore them in your model. That’s why regularization techniques are also called shrinkage techniques.

Regularization techniques are also called shrinkage techniques, because they shrink the value of the coefficients. Some coefficients can be shrunk to zero.

Even though it’s frequently used for linear models, regularization can also be activated to non-linear models.

The year of the (dog) nap

Now that you’re spending more time at home, you can absolutely see how much your dog naps!

The accepted doesn’t change much day-to-day, but you notice the nap continuance varies, depending on what they do or who they collaborate with.

Wouldn’t it absorbing if you could adumbrate how long your dog will nap tomorrow?

After giving it some thought, you can think of four main factors that affect your dog’s nap duration:

  • How many times they get dry treats,
  • How much playtime they have throughout the day,
  • If they see squirrels ambuscade around in the back yard,
  • If you got some bales delivered to your door.

Some of these activities and interactions may create anxiety, others just sheer excitement. Altogether they affect your dog’s energy levels and, consequently, how much they will nap during the day.

Since you want to adumbrate your dog’s nap duration, you start cerebration about it as a multivariate linear model.

The altered factors that affect nap continuance are the complete variables, while the nap continuance itself is the abased variable.

In apparatus acquirements terminology, the complete variables are features and the abased variable, what you want to predict, is the model target.

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Modeling your dog’s nap continuance as a multivariate linear model.

Looking at this nap continuance model, Beta 0 is the intercept, the value the target takes when all appearance are equal to zero.

The actual betas are the unknown coefficients which, along with the intercept, are the missing pieces of the model. You can beam the aftereffect of the aggregate of the altered features, but you don’t know all the capacity about how each affection impacts the target.

Once you actuate the value for each accessory you know the direction, either complete or negative, and the aftereffect of the impact each affection has in target.

With a linear model, you’re bold all appearance are complete of each other so, for instance, the fact that you got a commitment doesn’t have any impact on how many treats your dog gets in a day.

Additionally, you think there’s a linear accord amid the appearance and the target.

So, on the days you get to play more with your dog they’ll get more tired and will want to nap for longer. Or, on days when there are no squirrels alfresco your dog won’t need to nap as much, because they didn’t spend as much energy blockage alert and befitting an eye on the squirrels’ every move.

For how long will your dog nap tomorrow?

With the accepted idea of the model in your mind, you calm data for a few days. Now you have real observations of the appearance and the target of your model.

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A few days’ worth of observations of appearance and targets for your dog’s nap duration.

But there are still a few analytical pieces missing, the accessory values and the intercept.

One of the most accepted methods to find the coefficients of a linear model is Ordinary Least Squares.

The apriorism of Ordinary Least Squares (OLS) is that you’ll pick the coefficients that abbreviate the residual sum of squares, i.e., the total boxlike aberration amid your predictions and the empiric data[1].

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With the antithesis sum of squares, not all residuals are advised equally. You want to make an archetype of the times when the model generated predictions too far off from the empiric values.

It’s not so much about the anticipation being too far off above or below the empiric value, but the aftereffect of the error. You square the residuals and amerce the predictions that are too far off while making sure you’re only ambidextrous with complete values.

With antithesis sum of squares, it’s not so much about the anticipation being too far above or below the empiric value, but the aftereffect of that error.

This way when RSS is zero it really means anticipation and empiric values are equal, and it’s not just the bi-product of arithmetic.

In python, you can use ScikitLearn to fit a linear model to the data using Ordinary Least Squares.

Since you want to test the model with data it was not accomplished on, you want to hold out a allotment of your aboriginal dataset, into a test set. In this case, the test dataset sets aside 20% of the aboriginal dataset at random.

After applicable a linear model to the training set, you can check its characteristics.

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The coefficients and the ambush are the last pieces you needed to define your model and make predictions. The coefficients in the output array follow the order of the appearance in the dataset, so your model can be accounting as:

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It’s also useful to compute a few metrics to appraise the affection of the model.

R-squared, also called the accessory of determination, gives a sense of how good the model is at anecdotic the patterns in the training data, and has values alignment from 0 to 1. It shows how much of the airheadedness in the target is explained by the features[1].

For instance, if you’re applicable a linear model to the data but there’s no linear accord amid target and features, R-squared is going to be very close to zero.

Bias and about-face are metrics that help antithesis the two sources of error a model can have:

  • Bias relates to the training error, i.e., the error from predictions on the training set.
  • Variance relates to the generalization error, the error from predictions on the test set.

This linear model has a almost high variance. Let’s use regularization to reduce the about-face while trying to keep bias a low as possible.

Model regularization

Regularization is a set of techniques that advance a linear model in terms of:

  • Prediction accuracy, by abbreviation the about-face of the model’s predictions.
  • Interpretability, by shrinking or abbreviation to zero the coefficients that are not as accordant to the model[2].

With Ordinary Least Squares you want to abbreviate the Antithesis Sum of Squares (RSS).

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But, in a connected adaptation of Ordinary Least Squares, you want to shrink some of its coefficients to reduce all-embracing model variance. You do that by applying a amends to the Antithesis Sum of Squares[1].

In the regularized version of OLS, you’re trying to find the coefficients that minimize:

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The shrinkage penalty is the artefact of a tuning connected and corruption coefficients, so it will get abate as the corruption accessory allocation of the amends gets smaller. The tuning connected controls the impact of the shrinkage penalty in the antithesis sum of squares.

The shrinkage penalty is never activated to Beta 0, the intercept, because you only want to ascendancy the effect of the coefficients on the features, and the ambush doesn’t have a affection associated with it. If all appearance have accessory zero, the target will be equal to the value of the intercept.

There are two altered regularization techniques that can be activated to OLS:

  • Ridge Regression,
  • Lasso.

Ridge Regression

Ridge Corruption minimizes the sum of the square of the coefficients.

It’s also called L2 norm because, as the tuning parameter lambda increases the norm of the vector of least squares coefficients will always decrease.

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Even though it shrinks each model accessory in the same proportion, Ridge Corruption will never actually shrink them to zero.

The very aspect that makes this regularization more stable, is also one of its disadvantages. You end up abbreviation the model variance, but the model maintains its aboriginal level of complexity, since none of the coefficients were bargain to zero.

You can fit a model with Ridge Corruption by active the afterward code.

fit_model(features, targets, type='Ridge')

Here lambda, i.e., alpha in the scikit learn method, was arbitrarily set to 0.5, but in the next area you’ll go through the action of tuning this parameter.

Based on the output of the ridge regression, your dog’s nap continuance can be modeled as:

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Looking at other characteristics of the model, like R-squared, bias , and variance, you can see that all were bargain compared to the output of OLS.

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Ridge corruption was very able at shrinking the value of the coefficients and, as a consequence, the about-face of the model was decidedly reduced.

However, the complication and interpretability of the model remained the same. You still have four appearance that impact the continuance of your dog’s daily nap.

Let’s turn to Lasso and see how it performs.


Lasso is short for Least Complete Abbreviating and Alternative Operator [2], and it minimizes the sum of the complete values of the coefficients.

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It’s very agnate to Ridge corruption but, instead of the L2 norm, it uses the L1 norm as part of the shrinkage penalty. That’s why Lasso is also referred to as L1 regularization.

What’s able about Lasso is that it will actually shrink some of the coefficients to zero, thus abbreviation both about-face and model complexity.

Lasso uses a address called soft-thresholding[1]. It shrinks each accessory by a connected amount such that, when the accessory value is lower than the shrinkage constant it’s bargain to zero.

Again, with an arbitrary lambda of 0.5, you can fit lasso to the data.

In this case, you can see the feature squirrels was dropped from the model, because its accessory is zero.

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With Lasso, your dog’s nap continuance can be declared as a model with three features:

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Here the advantage over Ridge corruption is that you ended up with a model that is more interpretable, because it has fewer features.

Going from four to three appearance is not a big deal in terms of interpretability, but you can see how this could acutely useful in datasets that have hundreds of features!

Finding your optimal lambda

So far the lambda you used to see Ridge Corruption and Lasso in action was absolutely arbitrary. But there’s a way you can fine-tune the value of lambda to agreement that you can reduce the all-embracing model variance.

If you plot the root mean boxlike error against a connected set of lambda values, you can use the elbow technique to find the optimal value.

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Using the elbow address to find the optimal lambda value for Lasso and Ridge Regression.

This graph reinforces the fact that Ridge corruption is a much more stable address than Lasso. You can see the error starts off very high for the starting value of lambda = 0.01, but then it stabilizes right around 2.5. So, for Ridge Regression, a lambda of 2.5 would be the optimal value, since the error increases hardly after that.

As for Lasso, there’s a bit more variation. Zooming in to get more detail, the error absolutely starts off by accepting worse, when lambda is amid 0.15 and 0.2, before it stabilizes around lambda = 15.

Here’s how you can create these plots in Python.

We can verify this by applicable the data again, now with more targeted values.

You can affirm that for Lasso the about-face first gets worse but then gradually gets better and stabilizes around lambda equal to 15. At that point, Lasso dropped squirrels from the model and the all-embracing about-face is decidedly lower when compared than with lower values of lambda.

Using Lasso you ended abbreviation decidedly both about-face and bias.

With Ridge Corruption the model maintains all appearance and, as lambda increases, all-embracing bias and about-face gets lower. Like you noticed in the chart, when lambda is greater than 2.5 bias continues to get lower, but about-face absolutely gets worse.

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When to use Lasso vs Ridge?

Choosing the type of regularization address will depend on the characteristics of your model, as well as, the trade-offs you’re accommodating to make in terms of model accurateness and interpretability.

Use Lasso when …

Your model has a small number of appearance that stand out, i.e., have high coefficients, while the rest of the appearance have coefficients that are negligible.

In this case, Lasso will pick on the ascendant appearance and shrink the coefficients of the other appearance to zero.

Use Ridge Corruption when …

Your model has a lot of appearance and all have a almost similar weight in the model, i.e., their accessory values are very similar.


Depending on the botheration you’re alive on, it might be more useful to have a model that’s more interpretable compared to lower variance, yet overly complex, model. At the end of the day, it’s all about trade-offs!

Even though this was a very small dataset, the impact and capability of regularization were clear:

  • Coefficients and ambush were adjusted. Specifically, with Lasso, the feature squirrels could be alone from the model, because its accessory was shrunk to zero.
  • Variance was indeed reduced compared to the Ordinary Least Squares access when you picked the optimal lambda for the regularization.
  • You could see the bias-variance accommodation at play. A Ridge Corruption model with a lambda of 15, when compared to models with lower values of lambda, has a lower bias at the amount of accretion variance.

Appear February 23, 2021 — 14:00 UTC

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