Amongst a variety of items not quantitatively or statistically valued by buyers, Diamonds are possibly the most common. The purchase is far less from rational with a heavy bent on emotional ties. The jewelers would entice every man (and woman) by marketing it as a necessity for the occasion and as a status symbol, and by calling this pricey and unaffordable item as priceless. But what if you get the power to accurately estimate Diamond price before negotiating? Won’t that be a really cool edge!

However, the actual value of a diamond is determined by a gemologist after inspecting its various “features” (using proper machine learning basics terminology now since this article is on **Diamond price prediction using machine learning in python**) and applying a relative valuation principle of “compare and price”.

**CONSUMER’S GUIDE** to Buying Diamond

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## How to predict Diamond price using Machine learning & python

- Get the diamond price prediction dataset
- Data preprocessing
- Convert the features to log scale
- Machine Learning algorithms to predict diamond price
- Divide the data into independent variable features (X) and dependent variable (y)
- Train Test Split
- Linear regression for price prediction of diamonds
- Predict Diamond prices using K-nearest neighbors (KNN)
- Diamond price prediction using SVM (Support vector machine)
- Regression Evaluation Metrics
- Random Forest machine learning algorithm to predict diamond prices

- Diamond price prediction using neural networks
- Frequently Asked Questions (FAQ)

It’s holiday season! You can use machine learning regression for price prediction of the diamond you really desire — a little help in negotiating price

with local retailers.

This article is a no-brainer, easy-to-follow article to grasp machine learning basics in 15 mins for a Linear regression problem. The machine learning examples use diamond price prediction dataset with Python to show how to predict a number using minimal dataset at a fairly good accuracy.

In last 2 decades, the valuation and pricing has become more or less quantitative i.e. calculations based on values of many properties not just limiting to 4Cs (carat, cut, colour, clarity).

Properties like culet, pavilion, crown, girdle, girdle thickness polish, symmetry, fluorescence, table, depth and so on are the most easily identifiable and recordable features while the diamond is actually cut.

Tweet

Let me now start by showing you DIAMOND price prediction using machine learning & Python. In this article, I will show very simple steps of a machine learning process using a diamond price prediction dataset from Kaggle.

**Get the data**

The first steps in Machine learning process are to **Import all basic libraries**.

```
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
%matplotlib inline
from sklearn import preprocessing
```

Let’s begin training by using the diamond price dataset from Kaggle — it has basic data on more than 54000 diamonds. In a follow-up article, we will train the same machine learning model to predict diamond prices based on DIAMOND MARKET LIVE DATA from pricescope.com.

```
df = pd.read_csv(“diamonds.csv”)
df.drop(‘Unnamed: 0’, axis=1, inplace=True)
display(df.head(3))
```

The output is something like the following:

**Data preprocessing**

The next steps for diamond price prediction using Machine learning begin with basic data prepossessing. We will first check if any null values or unexpected data type are present in the Kaggle Diamond price prediction dataset to ensure accurate diamond price prediction using python.

`df.isnull().sum()df.info()`

Now we plot a Pair-plot of Price vs. 4 Cs (Carat, Cut, Color, Clarity) — the most popular and marketed properties of a diamond.

Read more about 4Cs at the following link:

https://www.diamonds.pro/education/4cs-diamonds/

```
# plot price vs. carat
sns.pairplot(df, x_vars=[’carat’], y_vars = [’price’])
# plot carat vs other Cs
sns.pairplot(df, x_vars=[’cut’, 'clarity’, 'color’], y_vars = [’carat’])
plt.show()
```

The output is something like the following:

We can see that the properties charts reveal a lot about how and where bulk of diamonds fall under each property value e.g. most bigger diamonds (higher carat) fall in **Fair** cut, **I1** clarity and **H-I** color. These are poor (commercial grade) diamonds generally sold by retail jewelry shops across the world to attract consumers with advertisements like ‘**Diamonds at 50% off**’

The price vs. carat chart also show that there are some outliers in the dataset i.e. few diamonds that are really *over priced*!

Now we need to see the distribution of the kaggle dataset for Diamond price prediction using Python. We will create a histogram plot for this. Being able to plot and visually interpret the models used in machine learning to predict diamond prices is a very vital part in completely grasping the motive of this article – learn Machine Learning basics.

Therefore, those steps in machine learning process needed for plotting and visually interpreting the data will be shown and explained everywhere in this article.

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First we define the `histplot()`

function.

```
def histplot(df, listvar):
fig, axes = plt.subplots(nrows=1, ncols=len(listvar), figsize=(20, 3))
counter=0
for ax in axes:
df.hist(column=listvar[counter], bins=20, ax=axes[counter])
plt.ylabel('Price')
plt.xlabel(listvar[counter])
counter = counter+1
plt.show()
```

Now we list the continuous variables and leave out the categorical variable. A continuous variable e.g. **carat** is one which has numerical values whereas a categorical variable is the one with alphanumeric values as categories e.g. **clarity**

```
linear_vars = df.select_dtypes(include=[np.number]).columns
display(list(linear_vars))
```

Output is `[‘carat’, ‘depth’, ‘table’, ‘price’, ‘l’, ‘w’, ‘d’]`

Now we plot the histogram

`histplot(df,linear_vars)`

This revels the distribution of each property. As expected, we see that the data is not normally distributed. After all, how can you expect a 1 carat diamond to be priced just at twice the price of a half-carat given all properties remain the same, while a 1 carat diamond looks much bigger to the eye when in a ring or earrings for that matter than a half carat one.

## Convert the features to log scale

One of the key steps in Machine learning process is to convert features having bimodal distribution to ones with BINOMIAL distribution.

### 1. Check for any ZERO value

Amongst features namely **table**, **depth**, **l**, **w**, **d, **check if any continuous variable has zero value. This results in a *division by zero* error when converted to log. Add a tiny number 0.01 to any zero value.

```
print(‘0 values →’, 0 in df.values)
df[linear_vars] = df[linear_vars] + 0.01
print(‘Filled all 0 values with 0.01. Now any 0 values? →’, 0 in df.values)
```

The output is:

```
0 values --> True
Filled all 0 values with 0.01. Now any 0 values? --> False
```

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### 2. View and remove outliers using z-score

Since we could briefly sense some outliers in the pairplot charts, lets dwell deeper and see whether there genuinely are any outliers. Let’s first begin by printing top X values of each diamond

```
def sorteddf(df, listvar):
for var in listvar:
display('sorted by ' + var + ' --> ' + str(list(df[listvar].sort_values(by=var,ascending=False)[var].head())))
```

Output is like:

```
sorted by carat --> [5.02, 4.51, 4.14, 4.02, 4.02]'
sorted by depth --> [79.01, 79.01, 78.21000000000001, 73.61, 72.91000000000001]'
sorted by table --> [95.01, 79.01, 76.01, 73.01, 73.01]'
sorted by price --> [21646.459999999995, 21640.709999999995, 21626.909999999996, 21624.609999999997, 21623.459999999995]'
sorted by l --> [10.75, 10.24, 10.15, 10.03, 10.02]'
sorted by w --> [58.91, 31.810000000000002, 10.549999999999999, 10.17, 10.11]'
sorted by d --> [31.810000000000002, 8.07, 6.99, 6.7299999999999995, 6.4399999999999995]'
```

From this list itself, we see that there are some outliers for *w,d*. Lets visualize those using boxplots. Create a boxplot function

```
def dfboxplot(df, listvar):
fig, axes = plt.subplots(nrows=1, ncols=len(listvar), figsize=(20, 3))
counter=0
for ax in axes:
df.boxplot(column=listvar[counter], ax=axes[counter])
plt.ylabel('Price')
plt.xlabel(listvar[counter])
counter = counter+1
plt.show()
```

Call dfboxplot to view outliers for all properties

`dfboxplot(df, linear_vars)`

We can now clearly visualize that there are outliers for *table*, *w* and *d* properties. Lets call `removeoutliers() `

function to remove the outliers based on z-score. There are several methods to remov outliers but I am going to follow the z-score process here since its the easiest to implement and delivers optimal results — after all, this is a no-brainer quickie article for newbie users.

For more on all other alogs to remove outliers, check these out:

Stackoverflow: Outliers in Pandas dataframe

```
def removeoutliers(df, listvars, z):
from scipy import stats
for var in listvars:
df1 = df[np.abs(stats.zscore(df[var])) < z]
return df1
df = removeoutliers(df, linear_vars,2)
```

Now, after calling dfboxbplot again to view outliers for all properties, we see the output as:

### 3. Convert to log scale

Since we saw earlier that most features (or properties of a diamond) are not normally distributed, and one of the most favored approach, if not a prerequisite, is to use Gaussian distributed (another name for normally distributed) data.

Quora: Gaussian vs. normal distribution

Medium: Gaussian-distribution-in-data-science

So in order to do perfect diamond price prediction using machine learning, we would need to *logarithmize* the values in the Kaggle diamond price prediction dataset.

```
# this log converts dataframe's features inplace
def convertfeatures2log(df, listvars):
for var in listvars:
df[var] = np.log(df[var])convertfeatures2log(df, linear_vars)
histplot(df, linear_vars)
```

The output now is:

## Convert categorical variable column to numerical column using labelencoder

We now have to convert all categorical columns to numerical columns using labelencoder. You may read more about labelencoder in the following webpage — quick and easy to understand description there.

Towards Data Science: encoding categorical features

First we define the `convert_catg() `

function to convert categorical columns to numerical columns

```
def convert_catg(df1):
from sklearn.preprocessing import LabelEncoder
le = LabelEncoder()
# Find the columns of object type along with their column index
object_cols = list(df1.select_dtypes(exclude=[np.number]).columns)
object_cols_ind = []
for col in object_cols:
object_cols_ind.append(df1.columns.get_loc(col))
# Encode the categorical columns with numbers
for i in object_cols_ind:
df1.iloc[:,i] = le.fit_transform(df1.iloc[:,i])
```

Next, we run the function and see the head of the dataframe.

`convert_catg(df)df.head(3)`

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Let’s now prepare our data further to start building the **Machine learning mode**l to predict diamond prices.

### Divide the data into independent variable features (X) and dependent variable (y)

The next step in our journey of diamond price prediction in python is to set the independent variable (X) and dependent variable (y).

X is the matrix (or DataFrame) for all the properties (independent features) and y is the vector for output (dependent variables) i.e. diamond price

```
X_df = df.drop([‘price’, ‘l’, ‘w’, ‘d’], axis=1)
y_df = df[[‘price’]] # two [[ to create a DF
```

Now, we determine correlations between price and all other attributes.

- I will be combining both X (already converted categorical to numerical) and y to form a new dataframe for correlation

```
df_le = X_df.copy()
# add a new column in dataframe — join 2 dataframe columns-wise
df_le[‘price’] = y_df[‘price’].values
df_le.corr()
```

The statement

df_le = X_dfmeans thatdf_lewill be like a pointer toX_df. Any change made todf_lewill actually be a change toX_df. Thereforedf_le = X_df.is better.copy()

- It seems diamond price is highly correlated with carat and fairly with table, color and clarity, but not much with cut

**Note on Feature scaling** — it seems its not needed here since we have already converted the diamond price prediction dataset properties into log. Nevertheless, if I had feature scaled, then the code would have been as written below:

```
from sklearn.preprocessing import StandardScaler
sc_X = StandardScaler()
X_df = sc_X.fit_transform(X_df)
X_df[0:3]
```

### Train Test Split

Data scientists generally split the dataset used for machine learning projects into either two or three subsets: 2 subsets for training and testing, while 3 for training, validation and testing. I will talk about it in detail a bit later. This data split prevents an algorithm from overfitting and underfitting. I have explained overfitting and underfitting briefly in my other post here:

The code for train_test_split is as follows:

```
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X_df, y_df, test_size=0.3, random_state=42)
```

Now I will take you through the steps needed to predict diamond price using machine learning and Python on SVM, KNN, Neural networks models.

Once you are really ready with your clean & corrected diamond price prediction dataset, coding a simple Machine Learning algorithm in python is a cakewalk.

Let me now demonstrate a few machine learning algorithms with code, output and charts.

An outline of what we will be doing in the next few pages is:

- split the diamond price prediction dataset into training set and test set.
- Train the algorithm on training set data
- Use the trained algorithm (or trained ML model) to estimate price from diamond properties in test data.
- Verify / visualize / measure the differences between the predicted prices and actual prices using scatterplots, histograms, accuracy metrics etc.

**Machine Learning algorithms**

Now is the time to learn regression models for diamond price prediction and start coding machine learning algorithms on this dataset.

After completing all necessary data pre-processing steps, let’s move ahead and see some ML algorithms in action.

### Linear regression

Let’s start with the simplest of all, the ubiquitous linear regression model to predict price of a diamond.

- Import
`LinearRegression`

class from**Sci-kit learn** - Create an object of LinearRegression model
- Fit the model to X_train and y_train
- Make predictions

```
# Import the class from Sci-kit learn
from sklearn.linear_model import LinearRegression
# Create an object of LinearRegression model
reg_all = LinearRegression()
# Fit the model to X_train and y_train
reg_all.fit(X_train,y_train)
# Make predictions
y_pred=reg_all.predict(X_test)
```

Now visualize the discrepancy of predictions vs. actual prices using scatterplot and histogram

```
import matplotlib.pyplot as plt
plt.scatter(y_test,y_pred)
```

```
import seaborn as sns
sns.distplot((y_test-y_pred),bins=50);
```

Note that this is a comparison between logarithm of actual prices from the diamond price prediction dataset, and the predictions from our linear regression ML model → See the code again. It is written as `plt.scatter(y_test,y_pred)`

after all features were converted to logarithm using `convertfeatures2log()`

This means to see actual discrepancies, we should un-logarithm it i.e. find the exponent of every price and prediction and then plot. The following code does this:

```
# convert prices and predictions back to exp
y_pred2 = np.exp(y_pred)
y_test2 = np.exp(y_test)
```

Now see the scatterplot and histogram again:

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### K-nearest neighbors (KNN)

Because so many API libraries exist for several machine learning algorithms, the code for all simple machine learning algorithms is straightforward.

On the lines of linear regression code, we use sklearn library for to predict diamond price using KNN or K-nearest neighbors.

```
from sklearn.neighbors import KNeighborsRegressor
reg_all = KNeighborsRegressor(n_neighbors = 8, metric = ‘minkowski’, p = 2)
reg_all.fit(X_train,y_train)
y_pred=reg_all.predict(X_test)
```

The distplot and scatterplot for **logarithmic **features are as follows. You can confirm from values in x-axis and y-axis that y_test and y_pred are log here.

The distplot and scatterplot for **absolute values **of features are as follows. You can confirm from values in x-axis and y-axis that y_test and y_pred are **NOT **log here.

### Support vector machines (SVM)

Let’s now look at diamond price prediction using SVM in Python.

- Import
`LinearRegression`

class from Sci-kit learn - Create an object of LinearRegression model
- Fit the model to X_train and y_train
- Make predictions

```
from sklearn.svm import SVR
regressor = SVR(kernel=’rbf’)
regressor.fit(X_train,y_train)
y_pred = regressor.predict(X_test)
```

Below is a comparison of scatterplots of log values of features (left plot) and absolute values (right plot) of features.

As of now, we can deduce that diamond price prediction using SVM is a better option because it gives a better metrics score and a better scatterplot than Linear Regression and KNN

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### Regression Evaluation Metrics

Here are three common evaluation metrics for any regression problem including our diamond price prediction using machine learning problem:

The code to call functions related to all 3 evaluation metrics is mentioned below.

```
from sklearn import metricsprint(‘MAE:’, metrics.mean_absolute_error(y_test, y_pred))
print(‘MSE:’, metrics.mean_squared_error(y_test, y_pred))
print(‘RMSE:’, np.sqrt(metrics.mean_squared_error(y_test, y_pred)))
# Output for SVM is :
MAE: 0.08815796872409032
MSE: 0.012811091991743056
RMSE: 0.11318609451581522
```

### Random Forest

Random forest is one of the most popular algorithms in most use cases / projects across industries. Its fast, easier to implement, needs lesser data, doesnt require extensive training and produces almost equally good results.

We will now use the sklearn code and kaggle dataset for Diamond price prediction using python with Random Forest machine learning algorithm.

```
from sklearn.ensemble import RandomForestRegressor
rf = RandomForestRegressor(n_estimators = 10)
rf.fit(X_train,y_train)
y_pred = rf.predict(X_test)
```

Now the metrics and their outputs

```
MAE: 0.08413000449351056
MSE: 0.012880789432555585
RMSE: 0.1134935655997977
```

It turns out that Random Forest is similar to the far slower SVM for diamond price prediction.

## Diamond price prediction using neural networks

Generally, neural networks or ANNs are more suited for classification problems requiring lots of complex logic / decision making and huge computations. They require larger datasets for their optimization to get the benefit of generalization and nonlinear mapping. But, if there’s not enough data, a plain regression model may be better suited despite a few nonlinearities.

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However, just for sake of completeness, I will show you the steps needed for diamond price prediction using Neural networks.

And for ANN, the best, fastest and easiest to use and code library is Keras. Read more about Keras at their official website:

### How to create an Artificial neural network in python, Keras & *kerasregressor*

The step-wise process to create a neural network to predict diamond prices is:

- Construct a baseline neural network in keras
- Compile and return a Keras model using
*KerasRegressor*, - Use the K-fold function to get perfect results
- Fit the estimator to diamond price prediction dataset (training data)
- Finally, predict the output (the diamond price predicted using neural networks)
- Evaluate the model using metrics between y-predicted vs. y-test

Now we go through each step in detail to create a neural network in keras.

Firstly import the libraries for our project:

```
from keras.models import Sequential
from keras.layers import Dense
from keras.optimizers import RMSprop
from keras.wrappers.scikit_learn import KerasRegressor
from sklearn.model_selection import cross_val_score
from sklearn.model_selection import KFold
```

Next, we construct a `baseline_model()`

function to create and return a Keras model

```
# define base model
def baseline_model():
# create model
model = Sequential()
# add 1st layer
model.add(Dense(output_dim=18, input_dim=11, activation='relu')) # kernel_initializer='normal',
# add hidden layer
model.add(Dense(output_dim=12, kernel_initializer='normal', activation='relu'))
# add output layer
model.add(Dense(1, kernel_initializer='normal'))
# Compile model
model.compile(loss='mean_squared_error', optimizer='adam')
return model
```

After designing the baseline neural network in keras, we run the `KerasRegressor()`

function.

- It returns the baseline Artificial neural network model built in Keras. It takes as input the model, epochs and batch-size.
- An
**epoch**defines the number of times the learning algorithm will work through the entire training dataset to update the weights for a neural network. An epoch that has one batch is called the batch gradient descent learning algorithm. For batch training, all the training samples pass through the learning algorithm simultaneously in one epoch before weights are updated. - The
**batch size**is a number of samples processed before the model is updated.

More information on `KerasRegressor `

can be found at Tensorflow website:

```
estimator = KerasRegressor(build_fn=baseline_model, epochs=10, batch_size=5)
kf = KFold(n_splits=5)
results = cross_val_score(estimator, X_train, y_train, cv=kf)
print(“Results: %.2f (%.2f) MSE” % (results.mean(), results.std()))
```

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Using the neural network built in keras, we run the `KFold()`

function.

K-Fold Cross Validation uses a given data set, splits it into a K number of folds where each fold is used as a testing set while other K-1 are used as training set.

- For example, for 10-Fold cross validation (K=10), the dataset is split into 10 folds.
- In the first iteration, the first fold is used for validation while the 9 remaining folds form the training set.
- In the second iteration, 2nd fold is used as the testing set while the rest serve as the training set.
- This process is repeated until each fold of the 10 folds have been used as testing sets.

Subsequently, the cross_val_score function takes the model, X and y, and kfold’s result as inputs and outputs multiple results — a list of regression model metrics scores. The `cross_val_score`

function splits the data, using `KFold`

as described above, into K pieces, trains on each combination of K-1 folds and gives back the metrics of the model.

Brief information and steps on

`KFold`

and`cross_val_score`

can be found in my other post; link below.

Finally, we fit the estimator to our training data to get predictions from our mini-project on diamond price prediction using neural networks.

```
estimator.fit(X_train, y_train)
y_pred = estimator.predict(X_test)# Plot a scatter plot like above to see prediction perfection
plt.scatter(y_test,y_pred)
```

The scatterplot output for log(features) is as below. It turns out that it is not as bad as we expected when I said that ANNs are mostly used for classification, not regression. You can yourself decide if you would like to invest resources in ANN deployment simply for diamond price prediction using machine learning.

**Top Data science News**

## References

For this particular post, I have referred to several websites including Beyond4Cs, Machine Learning Mastery and others few posts and websites on internet.

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The same post was first written on Medium.com. Please click here for part I and here for part II of my initial two posts on medium.

Thanks for reading this post. I have written several other such Machine Learning related articles on this website. If you liked this post, kindly comment and like using the comment form below.

## Frequently Asked Questions (FAQ)

### What are different types of Machine Learning?

This is one of the very first questions by readers of any guide on Machine Learning. Generally speaking,

There are 3 types of machine learning:

1. Supervised Learning, where an ML model makes predictions based on known or tagged or labeled data (data with tags or labels, thereby seemingly more meaningful) e.g. our diamond price prediction in python

2. Unsupervised Learning, where there’s no labeled data. An ML model here identifies patterns, relationships and anomalies to predict the outcome.

3. Reinforcement Learning, where the ML model for prediction learns continually based on the output its predicts and the rewards it receives for its previous actions.

### What is the process followed to create Machine learning models for prediction?

One of the first steps in Machine Learning process is to understand the business requirements and then identify the ML model. Then, a very simple 3-step machine learning basic process is followed to create ML models for prediction:

1. Train the model: Split the entire data to be used to predict diamond price into *train *and *test *data using train-test-split, or any other method. The train data is run on the agreed ML model for prediction. The model is tweaked regularly by editing model parameters unless accuracy, precision and recall and other tests match your expectations.

2. Test the model: Once the output is up to satisfactory levels, you use the test data to check if the model predicts with agreed accuracy. This would help determine if training is effective. On errors, change your ML model for prediction or retrain it with more data.

3. Deploy the model: Finally, upon several rounds of testing and training, once the model reaches your levels of expectations, you deploy the model into production.

### What is bias and variance?

**Bias **is the distance of predicted values (y-hat) from the actual values (y). Bias is High when the average predicted values are far from the actual values. **Variance **is high when the model performs good on the training set but not on test set. Variance shows how scattered are the predicted values from actual values.

### What can I get from a confusion matrix?

A Confusion matrix is used for evaluating the performance not for ML models for prediction but for classification. It is an 2 x 2 matrix which compares the actual values with those predicted by the machine learning model. It is the first step in machine learning basics on performance evaluation. Furthermore, it helps tell you how well the ML classification model is performing and the errors it is making.**True Positive (TP) **: The predicted value matches the actual value. E.g. actual value = positive, predicted value = positive**True Negative (TN): **The predicted value matches the actual value. E.g. actual value = *negative*, predicted value = *negative***False Positive (FP), also called Type 1 error**: Wrong prediction. E.g. actual value = negative , predicted value = positive**False Negative(FP), also called Type 2 error**: Wrong prediction. E.g. actual value = positive, predicted value = negative

### What is Semi-supervised Machine Learning?

Supervised learning uses labeled data while unsupervised learning uses no training data.

The third type, semi-supervised learning uses a small amount of labeled data and a large amount of unlabeled data.

### Precision and Recall formula in 10 seconds!

Another important item falling under Machine Learning basics.

Precision = (True Positive) / (True Positive + False Positive)

Recall = (True Positive) / (True Positive + False Negative)

Example of Precision & recall using one-class classification:

Precision means how many times you can “correctly recollect” the date when your girlfriend gave you the watch you are wearing.

Recall means how many times you can “recollect” that the watch was actually gifted by her, not by your Dad.

### What’s k-fold cross validation?

K-fold cross validation is a procedure used to estimate the skill of the model on unseen data.

K = number of groups your data is split into.**K-fold cross validation** procedure:

1. Shuffle the dataset randomly by splitting the dataset into k groups (10 groups)

2. For each unique group: (for i = 1 to 10)

a) Take the group as test data set (train_test_split = 10% test & 90% train) + remaining groups as a training data set

b) Fit a model on the training set and evaluate it on this test set

c) Retain the evaluation score and discard the model

Summarize the skill of the model using the sample of model evaluation scores

Hey Jatin,

This was very helpful. I took some of the steps and applied them to my own project and they were worth the effort. My RMSE and R2 scores improved. Keep publishing Data Science articles. I like walkthroughs like this. They help give my perspective on my own process.

Thank you. I’m glad you found it helpful.

Stay up the great work! You recognize, many individuals are looking around for this information, you

Thank you very much for the appreciation. I’m glad you liked the article.

Thank you very much. I’m glad you liked the article.