So far, you have explored what regression is with sample data gathered from the pumpkin pricing dataset that we will use throughout this lesson. You have also visualized it using Matplotlib.
So far you have explored what regression is with sample data gathered from the pumpkin pricing dataset that we will use throughout this lesson. You have also visualized it using Matplotlib.
Now you are ready to dive deeper into regression for ML. In this lesson, you will learn more about two types of regression: _basic linear regression_ and _polynomial regression_, along with some of the math underlying these techniques.
> Throughout this curriculum, we assume minimal knowledge of math, and seek to make it accessible for students coming from other fields, so watch for notes, callouts, diagrams, and other learning tools to aid in comprehension.
> Throughout this curriculum, we assume minimal knowledge of math, and seek to make it accessible for students coming from other fields, so watch for notes, 🧮 callouts, diagrams, and other learning tools to aid in comprehension.
## Prerequisite
### Prerequisite
You should be familiar by now with the structure of the pumpkin data that we are examining. You can find it preloaded and pre-cleaned in this lesson's _notebook.ipynb_ file. In the file, the pumpkin price is displayed per bushel in a new dataframe. Make sure you can run these notebooks in kernels in Visual Studio Code.
## Preparation
### Preparation
As a reminder, you are loading this data so to ask questions of it:
As a reminder, you are loading this data so as to ask questions of it.
- When is the best time to buy pumpkins?
- When is the best time to buy pumpkins?
- What price can I expect of a case of miniature pumpkins?
- Should I buy them in half-bushel baskets or by the 1 1/9 bushel box?
Let's keep digging into this data.
✅ In the previous lesson, you created a Pandas data frame and populated it with part of the original dataset, standardizing the pricing by the bushel. By doing that, however, you were _only_ able to gather about 400 data points and only for the fall months. Take a look at the data that we preloaded in this lesson's accompanying _notebook.ipynb_. The data is preloaded and an initial scatter plot is charted to show month data. Maybe we can get a little more detail about the nature of the data by cleaning it more.
In the previous lesson, you created a Pandas dataframe and populated it with part of the original dataset, standardizing the pricing by the bushel. By doing that, however, you were only able to gather about 400 datapoints and only for the fall months.
Take a look at the data that we preloaded in this lesson's accompanying notebook. The data is preloaded and an initial scatterplot is charted to show month data. Maybe we can get a little more detail about the nature of the data by cleaning it more.
## A linear regression line
As you learned in Lesson 1, the goal of a linear regression exercise is to be able to plot a line to:
- **Variable relationship**. Show the relationship between variables.
- **Make preditions**. Make accurate predictions on where a new data point would fall in relationship to that line.
### Understand the math
Since you'll use Scikit-learn, there's no reason to do this by hand (although you could!). In the main data-processing block of your lesson notebook, add a library from Scikit-learn to automatically convert all string data to numbers:
This line has an equation:
```text
Y = a + bX`.
```
It is typical of **Least-Squares Regression** to draw this type of line.
Todo infographic
If you look at the new_pumpkins dataframe now, you see that all the strings are now numeric. This makes it harder for you to read but much more intelligible for Scikit-learn!
```
Package Price
70 0 13.636364
71 0 16.363636
72 0 16.363636
73 0 15.454545
74 0 13.636364
... ... ...
1738 2 30.000000
1739 2 28.750000
1740 2 25.750000
1741 2 24.000000
1742 2 24.000000
415 rows × 2 columns
```
`X` is the 'explanatory variable'. `Y` is the 'dependent variable'.
The slope of the line is `b` and `a` is the y-intercept (where the line intersects with the Y-axis), which refers to the value of `Y` when `X = 0`.
In other words, and referring to our pumpkin data's original question: _predict the price of a pumpkin per bushel by month_:
- `X` would refer to the price.
- `Y` would refer to the month of sale.
The math that calculates the line must demonstrate the slope of the line, which is also dependent on the intercept, or where `Y` is situated when `X = 0`.
✅ You can observe the method of calculation for these values on the [Math is Fun](https://www.mathsisfun.com/data/least-squares-regression.html) web site.
### Least-squares regression
A common method of regression is **Least-Squares Regression**. Which means that all the data points, surrounding the regression line, are _squared_ and then added up.
Ideally, that final sum is as small as possible, because we want a low number of errors, or _least-squares_.
We want to model a line that has the least cumulative distance from all of our data points. We also square the terms before adding them since we are concerned with its magnitude rather than its direction.
- **Show variable relationships**. Show the relationship between variables
- **Make predictions**. Make accurate predictions on where a new datapoint would fall in relationship to that line.
It is typical of **Least-Squares Regression** to draw this type of line. The term 'least-squares' means that all the datapoints surrounding the regression line are squared and then added up. Ideally, that final sum is as small as possible, because we want a low number of errors, or `least-squares`.
### Correlation coefficent
We do so since we want to model a line that has the least cumulative distance from all of our data points. We also square the terms before adding them since we are concerned with its magnitude rather than its direction.
One more term to understand is the **Correlation Coefficient** between given `X` and `Y` variables.
> **🧮 Show me the math**
>
> This line can be expressed by an equation:
>
> ```
> Y = a + bX
> ```
>
> `X` is the 'explanatory variable'. `Y` is the 'dependent variable'. The slope of the line is `b` and `a` is the y-intercept, which refers to the value of `Y` when `X = 0`.
>
> todo infographic
>
> In other words, and referring to our pumpkin data's original question: "predict the price of a pumpkin per bushel by month", `X` would refer to the price and `Y` would refer to the month of sale.
>
> The math that calculates the line must demonstrate the slope of the line, which is also dependent on the intercept, or where `Y` is situated when `X = 0`.
>
> You can observe the method of calculation for these values on the [Math is Fun](https://www.mathsisfun.com/data/least-squares-regression.html) web site. Also visit [this Least-squares calculator](https://www.mathsisfun.com/data/least-squares-calculator.html) to watch how the numbers' values impact the line.
- **High correlation**. A plot with data points scattered in a neat line have high correlation.
- **Low correlation**. A plot with data points scattered everywhere between X and Y have a low correlation.
## Correlation
> A good linear regression model will be one that has a high (nearer to 1 than 0) Correlation Coefficient using the Least-Squares Regression method with a line of regression.
One more term to understand is the **Correlation Coefficient** between given X and Y variables. Using a scatterplot, you can quickly visualize this coefficient. A plot with datapoints scattered in a neat line have high correlation, but a plot with datapoints scattered everywhere between X and Y have a low correlation.
✅ Run the notebook accompanying this lesson and look at the City to Price scatter plot. Does the data associating City to Price for pumpkin sales seem to have high or low correlation, according to your visual interpretation of the scatter plot?
A good linear regression model will be one that has a high (nearer to 1 than 0) Correlation Coefficient using the Least-Squares Regression method with a line of regression.
## Exercise - create a Linear Regression Model correlating Pumpkin data points
✅ Run the notebook accompanying this lesson and look at the City to Price scatterplot. Does the data associating City to Price for pumpkin sales seem to have high or low correlation, according to your visual interpretation of the scatterplot?
Now that you have an understanding of the math behind this exercise the next step is create a Regression model to see if you can predict _which package of pumpkins will have the best pumpkin prices_.
✅ Someone buying pumpkins for a holiday pumpkin patch might want this information to be able to optimize their purchases of pumpkin packages for the patch.
## Prepare your data for regression
Since you'll use Scikit-Learn, there's no reason to do this by hand (although you could!).
Now that you have an understanding of the math behind this exercise, create a Regression model to see if you can predict which package of pumpkins will have the best pumpkin prices. Someone buying pumpkins for a holiday pumpkin patch might want this information to be able to optimize their purchases of pumpkin packages for the patch.
### Convert string data to numbers
In the main data-processing block of your lesson notebook, add a library from Scikit-Learn to automatically convert all string data to numbers.
Since you'll use Scikit-learn, there's no reason to do this by hand (although you could!). In the main data-processing block of your lesson notebook, add a library from Scikit-learn to automatically convert all string data to numbers:
If you look at the `new_pumpkins` dataframe now, you see that all the strings are now numeric. This makes it harder for you to read but much more intelligible for Scikit-Learn!
If you look at the new_pumpkins dataframe now, you see that all the strings are now numeric. This makes it harder for you to read but much more intelligible for Scikit-learn!
Now you can make more educated decisions (not just based on eyeballing a scatterplot) about the data that is best suited to regression.
Try to find a good correlation between two points of your data to potentially build a good predictive model.
Try to find a good correlation between two points of your data to potentially build a good predictive model. As it turns out, there's only weak correlation between the City and Price:
1. Let's see what the correlation is between `City` and `Price` and by using the method `corr()`:
1. Check the correlation between `Package` and `Price` next:
However there's a bit better correlation between the Package and its Price. That makes sense, right? Normally, the bigger the produce box, the higher the price.
`> 0.6`, that's better than the last thing we tried. There's a bit better correlation between the `Package` and its `Price`. That makes sense, right? Normally, the bigger the produce box, the higher the price.
A good question to ask of this data will be: 'What price can I expect of a given pumpkin package?'
✅ A good question to ask of this data will be: _What price can I expect of a given pumpkin package?_
Let's build this regression model
Let's build this regression model next.
## Building a linear model
## Exercise - building a linear model
Before building your model, do one more tidy-up of your data. Drop any null data and check once more what the data looks like.
Before building your model, do one more tidy-up of your data.
```python
new_pumpkins.dropna(inplace=True)
new_pumpkins.info()
```
1. Drop any null data and check once more what the data looks like.
Then, create a new dataframe from this minimal set and print it out:
```python
new_pumpkins.dropna(inplace=True)
new_pumpkins.info()
```
```python
new_columns = ['Package', 'Price']
lin_pumpkins = new_pumpkins.drop([c for c in new_pumpkins.columns if c not in new_columns], axis='columns')
1. Then, create a new dataframe from this minimal set and print it out:
lin_pumpkins
```
```python
new_columns = ['Package', 'Price']
lin_pumpkins = new_pumpkins.drop([c for c in new_pumpkins.columns if c not in new_columns], axis='columns')
lin_pumpkins
```
```output
Package Price
70 0 13.636364
71 0 16.363636
72 0 16.363636
73 0 15.454545
74 0 13.636364
... ... ...
1738 2 30.000000
1739 2 28.750000
1740 2 25.750000
1741 2 24.000000
1742 2 24.000000
415 rows × 2 columns
```
1. Now you can assign your X and y coordinate data:
```python
X = lin_pumpkins.values[:, :1]
y = lin_pumpkins.values[:, 1:2]
```
```python
X = lin_pumpkins.values[:, :1]
y = lin_pumpkins.values[:, 1:2]
```
✅ What's going on here? You're using [Python slice notation](https://stackoverflow.com/questions/509211/understanding-slice-notation/509295#509295) to create arrays to populate `X` and `y`.
> What's going on here? You're using [Python slice notation](https://stackoverflow.com/questions/509211/understanding-slice-notation/509295#509295) to create arrays to populate `X` and `y`.
2. Next, start the regression model-building routines:
1. Next, start the regression model-building routines:
```python
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_squared_error, mean_absolute_error
from sklearn.model_selection import train_test_split
```python
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score, mean_squared_error, mean_absolute_error
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
lin_reg = LinearRegression()
lin_reg.fit(X_train,y_train)
pred = lin_reg.predict(X_test)
accuracy_score = lin_reg.score(X_train,y_train)
print('Model Accuracy: ', accuracy_score)
```
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
lin_reg = LinearRegression()
lin_reg.fit(X_train,y_train)
Because the correlation isn't particularly good, the model produced isn't terribly accurate.
pred = lin_reg.predict(X_test)
```output
Model Accuracy: 0.3315342327998987
```
accuracy_score = lin_reg.score(X_train,y_train)
print('Model Accuracy: ', accuracy_score)
```
1. To visualize the line that's drawn in the process:
Because the correlation isn't particularly good, the model produced isn't terribly accurate.
```python
plt.scatter(X_test, y_test, color='black')
plt.plot(X_test, pred, color='blue', linewidth=3)
plt.xlabel('Package')
plt.ylabel('Price')
plt.show()
```
```output
Model Accuracy: 0.3315342327998987
```
3. You can visualize the line that's drawn in the process:
1. Test the model against a hypothetical variety:
```python
plt.scatter(X_test, y_test, color='black')
plt.plot(X_test, pred, color='blue', linewidth=3)
```python
lin_reg.predict( np.array([ [2.75] ]) )
```
plt.xlabel('Package')
plt.ylabel('Price')
The returned price for this mythological Variety is:
plt.show()
```
```output
array([[33.15655975]])
```
4. Test the model against a hypothetical variety:
That number makes sense, if the logic of the regression line holds true.
```python
lin_reg.predict( np.array([ [2.75] ]) )
```
The returned price for this mythological Variety is:
Congratulations, you just created a model that can help predict the price of a few varieties of pumpkins. Your holiday pumpkin patch will be beautiful. But you can probably create a better model!
```output
array([[33.15655975]])
```
That number makes sense, if the logic of the regression line holds true.
🎃 Congratulations, you just created a model that can help predict the price of a few varieties of pumpkins. Your holiday pumpkin patch will be beautiful. But you can probably create a better model!
## Polynomial regression
Another type of linear regression is polynomial regression. While sometimes there's a linear relationship between variables - the bigger the pumpkin in volume, the higher the price - sometimes these relationships can't be plotted as a plane or straight line.
@ -246,122 +203,114 @@ Take another look at the relationship between Variety to Price in the previous p
✅ Polynomials are mathematical expressions that might consist of one or more variables and coefficients
Polynomial regression creates a curved line to better fit nonlinear data.
Polynomial regression creates a curved line to better fit nonlinear data.
1. Let's recreate a dataframe populated with a segment of the original pumpkin data:
poly_pumpkins = new_pumpkins.drop([c for c in new_pumpkins.columns if c not in new_columns], axis='columns')
### Visualize in a chart
poly_pumpkins
```
A good way to visualize the correlations between data in dataframes is to display it in a 'coolwarm' chart:
1. Use the `Background_gradient()` method with `coolwarm` as argument value like so:
```python
corr = poly_pumpkins.corr()
corr.style.background_gradient(cmap='coolwarm')
```
This code creates a heat map looking like so:
2. Use the `Background_gradient()` method with `coolwarm` as its argument value:
Looking at this chart, you can visualize the good correlation between `Package` and `Price`. So you should be able to create a somewhat better model than the last one.
```python
corr = poly_pumpkins.corr()
corr.style.background_gradient(cmap='coolwarm')
```
This code creates a heatmap:
Looking at this chart, you can visualize the good correlation between Package and Price. So you should be able to create a somewhat better model than the last one.
### Create a pipeline
Scikit-learn includes a helpful API for building polynomial regression models - the `make_pipeline` [API](https://scikit-learn.org/stable/modules/generated/sklearn.pipeline.make_pipeline.html?highlight=pipeline#sklearn.pipeline.make_pipeline). A 'pipeline' is created which is a chain of estimators. In this case, the pipeline includes polynomial features, or predictions that form a nonlinear path.
1. Build out the X and y columns:
```python
X=poly_pumpkins.iloc[:,3:4].values
y=poly_pumpkins.iloc[:,4:5].values
```
1. Create the pipeline by calling the `make_pipeline()` method:
```python
from sklearn.preprocessing import PolynomialFeatures
You created a new data frame by calling `pd.DataFrame`. Furthermore you sorted the values by a call to `sort_values()`. Thereafter you created a polynomial plot that ended up looking like so:
You created a new dataframe by calling `pd.DataFrame`. Then you sorted the values by calling `sort_values()`. Finally you created a polynomial plot:
You can see a curved line that fits your data better.
1. Let's check the model's accuracy by calling `score()`:
You can see a curved line that fits your data better.
```python
accuracy_score = pipeline.score(X_train,y_train)
print('Model Accuracy: ', accuracy_score)
```
Let's check the model's accuracy:
And voila!
```python
accuracy_score = pipeline.score(X_train,y_train)
print('Model Accuracy: ', accuracy_score)
```
```output
Model Accuracy: 0.8537946517073784
```
And voila!
```output
Model Accuracy: 0.8537946517073784
```
That's better! Try to predict a price:
### Do a prediction
Let's see where we are, can we input a new value and get a prediction?
1. Call `predict()` to make a prediction:
```python
pipeline.predict( np.array([ [2.75] ]) )
```
Can we input a new value and get a prediction?
You are given this prediction:
Call `predict()` to make a prediction:
```python
pipeline.predict( np.array([ [2.75] ]) )
```
You are given this prediction:
```output
array([[46.34509342]])
```
```output
array([[46.34509342]])
```
It does make sense, if you compare it to the polynomial plot! And, if this is a better model than the previous one, looking at the same data, you need to budget for these more expensive pumpkins!
It does make sense, given the plot! And, if this is a better model than the previous one, looking at the same data, you need to budget for these more expensive pumpkins!
🏆 Well done! You created two regression models in one lesson. In the final section on regression, you will learn about logistic regression to determine categories.
---
## 🚀Challenge
Test several different variables in this notebook to see how correlation corresponds to model accuracy.
@ -374,4 +323,4 @@ In this lesson we learned about Linear Regression. There are other important typ
Jen Looper
4 years ago