R Losses Regression

R Losses Regression Rating: 7,7/10 859 votes

In a regression problem, we aim to predict the output of a continuous value, like a price or a probability. Contrast this with a classification problem, where we aim to predict a discrete label (for example, where a picture contains an apple or an orange).

Loess Regression Lines R
R Losses Regression Formula
R Loess Regression Ggplot
R Losses Regression Analysis
R Loess Regression

For a discussion of various pseudo-R-squareds see Long and Freese (2006) or our FAQ page What are pseudo R-squareds? Diagnostics: The diagnostics for logistic regression are different from those for OLS regression. For a discussion of model diagnostics for logistic regression, see Hosmer and Lemeshow (2000, Chapter 5). Bias-Variance Trade-Off in Multiple Regression. Let's kick off with the basics: the simple.

This notebook builds a model to predict the median price of homes in a Boston suburb during the mid-1970s. To do this, we’ll provide the model with some data points about the suburb, such as the crime rate and the local property tax rate.

Formula: a formula specifying the numeric response and one to four numeric predictors (best specified via an interaction, but can also be specified additively). Will be coerced to a formula if necessary. Data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are. LiblineaR can produce 10 types of (generalized) linear models, by combining several types of loss functions and regularization schemes. The regularization can be L1 or L2, and the losses can be the regular L2-loss for SVM (hinge loss), L1-loss for SVM, or the logistic loss for logistic regression. The default value for type is 0. See details below. In this post I will provide R code that implement’s the combination of repeated running quantile with the LOESS smoother to create a type of “quantile LOESS” (e.g: “Local Quantile Regression”). This method is useful when the need arise to fit robust and resistant (Need to be verified) a smoothed line for a quantile (an Continue reading 'Quantile LOESS – Combining a moving.

The Boston Housing Prices dataset

The Boston Housing Prices dataset is accessible directly from keras.

Examples and features

This dataset is much smaller than the others we’ve worked with so far: it has 506 total examples that are split between 404 training examples and 102 test examples:

The dataset contains 13 different features:

Per capita crime rate.
The proportion of residential land zoned for lots over 25,000 square feet.
The proportion of non-retail business acres per town.
Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
Nitric oxides concentration (parts per 10 million).
The average number of rooms per dwelling.
The proportion of owner-occupied units built before 1940.
Weighted distances to five Boston employment centers.
Index of accessibility to radial highways.
Full-value property-tax rate per $10,000.
Pupil-teacher ratio by town.
1000 * (Bk - 0.63) ** 2 where Bk is the proportion of Black people by town.
Percentage lower status of the population.

Each one of these input data features is stored using a different scale. Some features are represented by a proportion between 0 and 1, other features are ranges between 1 and 12, some are ranges between 0 and 100, and so on.

Let’s add column names for better data inspection.

Labels

The labels are the house prices in thousands of dollars. (You may notice the mid-1970s prices.)

Normalize features

It’s recommended to normalize features that use different scales and ranges. Although the model might converge without feature normalization, it makes training more difficult, and it makes the resulting model more dependent on the choice of units used in the input.

We are going to use the feature_spec interface implemented in the tfdatasets package for normalization. The feature_columns interface allows for other common pre-processing operations on tabular data.

The spec created with tfdatasets can be used together with layer_dense_features to perform pre-processing directly in the TensorFlow graph.

We can take a look at the output of a dense-features layer created by this spec:

Note that this returns a matrix (in the sense that it’s a 2-dimensional Tensor) withscaled values.

Create the model

Let’s build our model. Here we will use the Keras functional API - which is the recommended way when using the feature_spec API. Note that we only need to pass the dense_features from the spec we just created.

We then compile the model with:

We will wrap the model building code into a function in order to be able to reuse it for different experiments. Remember that Keras fit modifies the model in-place.

Train the model

The model is trained for 500 epochs, recording training and validation accuracy in a keras_training_history object.We also show how to use a custom callback, replacing the default training output by a single dot per epoch.

Now, we visualize the model’s training progress using the metrics stored in the history variable. We want to use this data to determine how long to train before the model stops making progress.

This graph shows little improvement in the model after about 200 epochs. Let’s update the fit method to automatically stop training when the validation score doesn’t improve. We’ll use a callback that tests a training condition for every epoch. If a set amount of epochs elapses without showing improvement, it automatically stops the training.

The graph shows the average error is about $2,500 dollars. Is this good? Well, $2,500 is not an insignificant amount when some of the labels are only $15,000.

Let’s see how did the model performs on the test set:

Predict

Finally, predict some housing prices using data in the testing set:

Conclusion

This notebook introduced a few techniques to handle a regression problem.

Loess Regression Lines R

Mean Squared Error (MSE) is a common loss function used for regression problems (different than classification problems).
Similarly, evaluation metrics used for regression differ from classification. A common regression metric is Mean Absolute Error (MAE).
When input data features have values with different ranges, each feature should be scaled independently.
If there is not much training data, prefer a small network with few hidden layers to avoid overfitting.
Early stopping is a useful technique to prevent overfitting.

loess {stats}

R Documentation

Local Polynomial Regression Fitting

Description

Fit a polynomial surface determined by one or more numericalpredictors, using local fitting.

Usage

Arguments

`formula`	a formula specifying the numeric response andone to four numeric predictors (best specified via an interaction,but can also be specified additively). Will be coerced to a formulaif necessary.
`data`	an optional data frame, list or environment (or objectcoercible by `as.data.frame` to a data frame) containingthe variables in the model. If not found in `data`, thevariables are taken from `environment(formula)`,typically the environment from which `loess` is called.
`weights`	optional weights for each case.
`subset`	an optional specification of a subset of the data to beused.
`na.action`	the action to be taken with missing values in theresponse or predictors. The default is given by`getOption('na.action')`.
`model`	should the model frame be returned?
`span`	the parameter α which controls the degree ofsmoothing.
`enp.target`	an alternative way to specify `span`, as theapproximate equivalent number of parameters to be used.
`degree`	the degree of the polynomials to be used, normally 1 or2. (Degree 0 is also allowed, but see the ‘Note’.)
`parametric`	should any terms be fitted globally rather thanlocally? Terms can be specified by name, number or as a logicalvector of the same length as the number of predictors.
`drop.square`	for fits with more than one predictor and`degree = 2`, should the quadratic term be dropped for particularpredictors? Terms are specified in the same way as for`parametric`.
`normalize`	should the predictors be normalized to a common scaleif there is more than one? The normalization used is to set the10% trimmed standard deviation to one. Set to false for spatialcoordinate predictors and others known to be on a common scale.
`family`	if `'gaussian'` fitting is by least-squares, and if`'symmetric'` a re-descending M estimator is used with Tukey'sbiweight function. Can be abbreviated.
`method`	fit the model or just extract the model frame. Can be abbreviated.
`control`	control parameters: see `loess.control`.
`...`	control parameters can also be supplied directly(if`control` is not specified).

Details

Fitting is done locally. That is, for the fit at point x, thefit is made using points in a neighbourhood of x, weighted bytheir distance from x (with differences in ‘parametric’variables being ignored when computing the distance). The size of theneighbourhood is controlled by α (set by span orenp.target). For α < 1, theneighbourhood includes proportion α of the points,and these have tricubic weighting (proportional to (1 - (dist/maxdist)^3)^3). Forα > 1, all points are used, with the‘maximum distance’ assumed to be α^(1/p)times the actual maximum distance for p explanatory variables.

For the default family, fitting is by (weighted) least squares. Forfamily='symmetric' a few iterations of an M-estimationprocedure with Tukey's biweight are used. Be aware that as the initialvalue is the least-squares fit, this need not be a very resistant fit.

It can be important to tune the control list to achieve acceptablespeed. See loess.control for details.

Value

An object of class 'loess'.

Note

As this is based on cloess, it is similar to but not identical tothe loess function of S. In particular, conditioning is notimplemented.

The memory usage of this implementation of loess is roughlyquadratic in the number of points, with 1000 points taking about 10Mb.

degree = 0, local constant fitting, is allowed in thisimplementation but not documented in the reference. It seems very littletested, so use with caution.