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https://github.com/fedesgh/ann_solving_heteroscedasticity_regression_for_house_price_using_keras_tuner

Trying to solve heteroscedasticity using transformations, and differents models.
https://github.com/fedesgh/ann_solving_heteroscedasticity_regression_for_house_price_using_keras_tuner

ann boxcox keras-tuner kmeans ols wls

Last synced: 5 days ago
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Trying to solve heteroscedasticity using transformations, and differents models.

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README 

        
Using a dataset of house prices from (https://www.kaggle.com/datasets/harlfoxem/housesalesprediction)

We aim to build an artificial neural network (ANN) optimized with Keras-Tuner.



♦The first model shows signs of **heteroscedasticity** then we tried several steps to improve the metrics and the understanding of the data set.



♦About the data set: 



Number of rows: 21613



Number of columns: 21



We use 18 columns as features (inputs), 1 as target (continuous variable "price"), and drop 2



## First model ANN



♦Hyperparameters:



num_layers: 4



units_0: 243



activation: relu



dropout: False



lr: 0.00038877295727384804



units_1: 441



units_2: 234



units_3: 342



units_4: 252



units_5: 63



units_6: 45



Score: 72580.02265625



♦Metrics: 



MAE:  71293.14455383993



MAPE:  0.12926294242315003



RMSE:  129690.04519255627



![images/images_ann/metrics.png](images/images_ann/metric.png)



## Breusch-Pagan Test



♦ The results show signs of **heteroscedesticity** so we perform a **Breusch-Pagan Test** (WLS.ipynb) wich confirms it (**p-value = 0**).



## Trying to solve **heteroscedesticity**. 



There are several ways to overcome **heteroscedesticity**, one of them is to try to transform the target. 



### ANN with Box-Cox transformation.



We transform the target with **Box-Cox** (ANN_regression_Box_Cox.ipynb) and store its **lambda** value. After training the **ANN** we detransform using the inverse **Box-Cox** function with **lambda** the predicted values and obtain the metrics



♦Metrics:



MAE:  76957.75244332664



MAPE:  0.14025944529697412



RMSE:  141284.0378744411



![images/images_box/metricfinal.png](images/images_box/metricfinal.png)



![images/images_box/res.png](images/images_box/res.png)



### WLS model.



Another solution may be to use a weighted least squares (WLS) model that we developed (WLS.ipybn) using the **statsmodels** library.



First we need to get **MSE** from the trained **OLS** model so that we can use it to assign **weights** as parameter in **WLS**.



♦Metrics:



![images/imageswls/metrics.jpg](images/imageswls/metrics.jpg)



![images/imageswls/metrics2.jpg](images/imageswls/metrics2.jpg)



OlS MAE:  125881.5718112254



OlS MAPE:  0.25561092052451



OlS RMSE:  201188.16048555568



==================



WLS MAE:  118120.09129246318



WLS MAPE:  0.21997946214696784



WLS MAPE:  217696.18526827247



♦Results: 



Although **WLS** performs better than **OLS** as it is supposed to do in this case, both models perform poorly compared to our **ANN** with MAE: 72580.02265625.



## Segmentation through Kmeans 



The idea is to build an unsupervised model (Segmentation1.ipybn) to segment the data before applying a regression model to predict the price. Hopefully, we will be able to differentiate complicated data sets from simple ones. We train an optimal Kmeans with **K=13** which shows us that there are two groups of data (3,6) with a wider price range.



![images/images_seg/cluster.png](images/images_seg/cluster.png)



![images/Segmentation/Seglabels.jpg](images/images_seg/labesstats.jpg)



Then we separate the data in labels **[3,6]** and **[0,1,2,4,5,7,8,9,10,11,12]** and build an **ANN** for each group 



### ANN for labels **[0,1,2,4,5,7,8,9,10,11,12]** (labels != 3 or 6): 



♦Metrics:



![images/images_seg/metrics_no63.png](images/images_seg/metrics_no63.png)



MAE:  61773.60449419276



MAPE:  0.1276305452895651



RMSE:  102346.97434221503



### ANN for labels [3,6]



♦Metrics:



![images/images_seg/metricssi63.png](images/images_seg/metricssi63.png)



MAE:  189728.11892564403



MAPE:  0.14515582486390521



RMSE:  320830.6859570142



## Quantile attempt 



We treat **heteroscedasticity** as **outliers**



First sort the data and graph .95 quantile:



![images/images_quant/quantile.png](images/images_quant/quantile.png)



### Build an ANN classifier to detect whether new data belongs outside (1) or inside (0) the 0.95 quantile



♦Structure: 



Loss: binary_crossentropy



num_layers: 16



units_0: 54



activation: relu



dropout: False



lr: 0.00010406284012562844



units_1: 234



units_2: 135



units_3: 261



units_4: 180



units_5: 261



units_6: 189



units_7: 81



units_8: 243



units_9: 36



units_10: 396



units_11: 18



units_12: 18



units_13: 18



units_14: 18



units_15: 18



Score: 0.04392734244465828



♦Metrics



![images/images_quant/metrics.png](images/images_quant/metrics.png)



### Training ANN regression for .95 quantile



♦Metrics:



![images/images_quant/train95.png](images/images_quant/train95.png)



MAE:  65967.06092114073



MAPE:  0.14311744919508246



RMSE:  96768.70049794603



### Training an ANN regression for .05 



♦Metrics:



![images/images_quant/train05.png](images/images_quant/train05.png)



MAE:  303073.26987327187



MAPE:  0.1882879259314524



RMSE:  458908.4186139931



### Final and most important step:



We classify all the data through our **ANN classifier** to predict whether they belong to the .95 quantile or not. 



After classification we apply the correct **ANN regression** for their classes



**Notice that there will be misclassified instances and we are interested in how the models will perform**



♦Results: 



**For data classified as 0.95 quantile**: 



![images/images_quant/finalmetric.95.png](images/images_quant/finalmetric.95.png)



MAE:  42503.68199954071



MAPE:  0.09253917253987413



RMSE:  72454.40221470839



**For data classified as 0.05**:



![images/images_quant/finalmetric05.png](images/images_quant/finalmetric05.png)



MAE:  198023.983157277



MAPE:  0.12319025379957788



RMSE:  339525.5425219687



## Foot note



I noticed that there is an issue with the WLS formula in the statsmodels documentation.(https://www.statsmodels.org/dev/generated/statsmodels.regression.linear_model.WLS.html)



It is not clear whether we should apply **w^2** or **w**, the first option works better in our **WLS**



There is an open forum on the case:



(https://stackoverflow.com/questions/46000839/is-the-example-of-wls-in-statsmodels-wrong)