This is part two of the series. In part one, we used linear regression model to predict the prices of used Toyota Corollas. There are some overlap in the materials for those just reading this post for the first time. For those who read the part 1 of the series using linear regression, then you can safely skip to the section where I applied neural networks to the same data set.
In this post, we will use neural networks! Skip to the Nueral Network analysis section if you’ve read part 1 of this series.
Let’s load in the Toyota Corolla file and check out the first 5 lines to see what the data set looks like:
## Price Age KM FuelType HP MetColor Automatic CC Doors Weight
## 1 13500 23 46986 Diesel 90 1 0 2000 3 1165
## 2 13750 23 72937 Diesel 90 1 0 2000 3 1165
## 3 13950 24 41711 Diesel 90 1 0 2000 3 1165
## 4 14950 26 48000 Diesel 90 0 0 2000 3 1165
## 5 13750 30 38500 Diesel 90 0 0 2000 3 1170
Price, Age, KM(kilometers driven), Fuel Type, HP(horsepower), Automatic or Manual, Number of Doors, and Weight in pounds are the data collected in this file for Toyota Corollas.
In predictive models, there is a response variable(also called dependent variable), which is the variable that we are interested in predicting.
The independent variables(the predictors) are one or more numeric variables we are using to predict the response variable. Given we are using a linear regression model, we are assuming the relationship between the independent and dependent variables follow a straight line. Here. with neural network, we DO NOT assume a linear relationship. In fact, that’s part of the power and flexibility of a neural network is that it can model nonlinearities in data very well.
But before we start our modeling exercise, it’s good to take a visual look at what we are trying to predict to see what it looks like. Since we are trying to predict Toyota Corolla prices with historical data, let’s do a simple histogram plot to see the distribution of Corolla prices:
One of the main steps in the predictive analytics is data transformation. Data is never in the way you want them. One might have to do some kinds of transformations to get it to the way we need them either because the data is dirty, not of the type we want, out of bounds, and a host of other reasons.
In this case, we need to convert the categorical variables to numeric variables to feed into our linear regression model, because linear regression models only take numeric variables.
The categorical variable we want to do the transformation on is Fuel Types. We that there are 3 Fuel Types: 1) CNG 2) Diesel 3) Petrol
summary(corolla$FuelType)
## CNG Diesel Petrol
## 17 155 1264
So, we can convert the categorical variable Fuel Type to two numeric variables: FuelType1 and FuelType2. We assign CNG to a new variable FuelType1 in which a 1 represents it’s a CNG vehicle and 0 it’s not. Likewise, we assign Diesel to a new variable FuelType2 in which a 1 represents it’s a Diesel vehicle and 0 it’s not.
So, what do we do with PETROL vehicles? This is represented by the case when BOTH FuelType1 and FuelType2 are zero.
## Price Age KM HP MetColor Automatic CC Doors Weight FuelType1
## 1 13500 23 46986 90 1 0 2000 3 1165 0
## 2 13750 23 72937 90 1 0 2000 3 1165 0
## 3 13950 24 41711 90 1 0 2000 3 1165 0
## FuelType2
## 1 1
## 2 1
## 3 1
The next step in predictive analytics is to explore our underlying. Let’s do a few plots of our explanatory variables to see how they look against Price.
This plot is telling and fits out intuition. The newer the car the more expensive it is.
More such plots (comparing price with various variables) can be found in the original article.
Now, it’s generally NOT a good idea to use your ENTIRE data sample to fit the model. What we want to do is to train the model on a sample of the data. Then we’ll see how it perform outside of our training sample. This breaking up of our data set to training and test set is to evaluate the performance of our models with unseen data. Using the entire data set to build a model then using the entire data set to evaluate how good a model does is a bit of cheating or careless analytics.
We use the first 1000 rows of data as training sample.
We have to normalized the data, which in this case means fitting into the range from 0 to 1. This is for theoretical reasons that will be explained in detail in a follow-up blog. Let’s take a quick peek at the normalized data.
## Price Age KM HP MetColor Automatic CC
## 382 0.1275797 0.6708861 0.7166202 0.02439024 1 0 1.0000000
## 534 0.2831144 0.6455696 0.1962477 0.33333333 0 0 0.4285714
## 822 0.1538462 0.7974684 0.2885403 0.33333333 1 0 0.4285714
## 1302 0.0956848 1.0000000 0.2919271 0.33333333 1 1 0.4285714
## 289 0.2831144 0.5443038 0.1819637 0.22764228 1 0 0.1428571
## Doors Weight FuelType1 FuelType2
## 382 0.6666667 0.16260163 0 1
## 534 1.0000000 0.12195122 0 0
## 822 0.6666667 0.05691057 0 0
## 1302 1.0000000 0.16260163 0 0
## 289 1.0000000 0.09756098 0 0
Now, let’s feed the normalized data into our neural network
## hidden: 10, 10, 10 thresh: 0.01 rep: 1/1 steps: 1000 min thresh: 0.01444780388
## 1914 error: 0.68337 aic: 683.36673 bic: 2356.91128 time: 4.79 secs
Let’s see what it looks like(looks like a complex brain with all its neural connections):
We are using neural network with 3 hidden layers(denoted by Hs) and each hidden layer has 10 neurons. The Bs are biases introduced.
In a follow-up blog, I’ll explore the theory behind neural networks and it will make it clear how the math works.
Of course, we see all of our independent variables as inputs(Is) and one output layer which is the Price of used Toyota Corollas.
The real test of a good model is to test the model with data that it has not fitted. Here’s where the rubber meets the road. We apply our model to unseen data to see how it performs.
Let’s feed the test data(unseen) to our neural network.
But first we also have to normalized the test data set as well. Here are the first 5 rows:
## Age KM HP MetColor Automatic CC
## 3 0.2820512821 0.20138764829 0.1707317073 1 0 1.0000000000
## 11 0.2948717949 0.15189775630 1.0000000000 0 0 0.7142857143
## 14 0.3717948718 0.11104566106 1.0000000000 1 0 0.7142857143
## 15 0.3846153846 0.16478926963 1.0000000000 1 0 0.7142857143
## 16 0.3333333333 0.09047235084 1.0000000000 0 0 0.7142857143
## Doors Weight FuelType1 FuelType2
## 3 0 0.3437500000 0 1
## 11 0 0.3854166667 0 0
## 14 0 0.3854166667 0 0
## 15 0 0.3854166667 0 0
## 16 0 0.3854166667 0 0
Once we normalized the test data, let’s feed it into the neural network and see what the predictions are. Here are the first 5 rows of the normalized predictions:
## [,1]
## 3 0.4509980742
## 11 0.5642686175
## 14 0.5539187028
## 15 0.5105490069
## 16 0.5683168430
Once we have the predictions, we have to denormalized it to get it back to the car Prices!
Here are the first 5 rows of the Actual Prices vs. the Predictioned Prices:
## actual Price
## 3 13950 16369.09868
## 11 20950 19387.75866
## 14 21500 19111.93343
## 15 22500 17956.13103
## 16 22000 19495.64387
Here are some common metrics to see how well the model predicts using various error metrics. The main takeway is we want our forecast errors to be small as possible. The smaller the forecast error the better the model is at predicting unseen data.
me
# mean error
## [1] -425.1604158
ME is the mean error. The ideal ME is zero, which means on average the predicted value perfectly matches the actual value. This is rarely if ever the case. As in all things, we must determine what is an acceptable level of errors for our predictive analytics model and accept it. No such thing as a perfect model with zero forecast error.
rmse
# root mean square error
## [1] 1646.459233
RMSE is root mean squared error. A mean squared error(MSE) is the average of the sauared differences between the predicted value and the actual value. The reason we square is to not account for sign differences(negative differences and positive differences are the same thing when squared). RMSE brings it back to our normal unit by taking the square root of MSE>
mape
# mean absolute percent error## [1] 11.02359326
MAPE stands for mean absoute percent error and express the forecast errors in percentages.
On average, our model had a forecast error of only 11%. Not bad for a first pass at this data set using neural network. With neural network, we usually need more features engineering than the linear regression. Here, we did no features engineering to see how it would perform. And it performs very similar to the linear regresion. Are there ways to improve the model performances? Yes, to both linear regression and neural networks. We will cover those topics in future blog posts.
Hope you enjoyed this and are excited in applying predictive analytics models to your problem space.
In follow on blogs I’ll explain in further details the theories behind these methods and the differences and similarities between the two.
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Comment
Hi Peter, I realise it's a while since you wrote this article and part 2 but is it possible to let me have the R codes for these pages?
I have managed to create the R code for tPart 1 regression but as for the rest, I would appreciate your hellp.
Best wishes
Dunccan
Posted 12 April 2021
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