Linear Regressions

A Regression is a method to determine the relationship between one variable (y) and other variables (x).

In statistics, a Linear Regression is an approach to modeling a linear relationship between y and x.

In Machine Learning, a Linear Regression is a supervised machine learning algorithm.

Scatter Plot

This is the scatter plot (from the previous chapter):

Example

<!DOCTYPE html>
<html>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
<body>

// Define Data
const data = [{
x:xArray,
y:yArray,
mode:"markers"
}];

// Define Layout
const layout = {
xaxis: {range: [40, 160], title: "Square Meters"},
yaxis: {range: [5, 16], title: "Price in Millions"},
title: "House Prices vs. Size"
};

// Display using Plotly
Plotly.newPlot("myPlot", data, layout);
</script>

</body>
</html>

Predicting Values

From the scattered data above, how can we predict future prices?

Use hand drawn linear graph
Model a linear relationship
Model a linear regression

Linear Graphs

This is a linear graph predicting prices based on the lowest and the highest price:

Example

<!DOCTYPE html>
<html>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
<body>

const data = [
{x:xArray, y:yArray, mode:"markers"},
{x:[50,150], y:[7,15], mode:"line"},
];

const layout = {
xaxis: {range: [40, 160], title: "Square Meters"},
yaxis: {range: [5, 16], title: "Price in Millions"},
title: "House Prices vs. Size"
};

Plotly.newPlot("myPlot", data, layout);
</script>

</body>
</html>

From a Previous Chapter

A linear graph can be written as y = ax + b

Where:

y is the price we want to predict
a is the slope of the line
x are the input values
b is the intercept

Linear Relationships

This Model predicts prices using a linear relationship between price and size:

Example

<!DOCTYPE html>
<html>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
<body>

// Calculate Slope
let xSum = xArray.reduce(function(a, b){return a + b;}, 0);
let ySum = yArray.reduce(function(a, b){return a + b;}, 0);
let slope = ySum / xSum;

// Generate values
const xValues = [];
const yValues = [];
for (let x = 50; x <= 150; x += 1) {
xValues.push(x);
yValues.push(x * slope);
}

const data = [
{x:xArray, y:yArray, mode:"markers"},
{x:xValues, y:yValues, mode:"line"}
];

const layout = {
xaxis: {range: [40, 160], title: "Square Meters"},
yaxis: {range: [5, 16], title: "Price in Millions"},
title: "House Prices vs. Size"
};

Plotly.newPlot("myPlot", data, layout);
</script>

</body>
</html>

In the example above, the slope is a calculated average and the intercept = 0.

Using a Linear Regression Function

This Model predicts prices using a linear regression function:

Example

<!DOCTYPE html>
<html>
<script src="https://cdn.plot.ly/plotly-latest.min.js"></script>
<body>

// Calculate Sums
let xSum=0, ySum=0, xxSum=0, xySum=0;
let count = xArray.length;
for (let i = 0, len = count; i < count; i++) {
xSum += xArray[i];
ySum += yArray[i];
xxSum += xArray[i] * xArray[i];
xySum += xArray[i] * yArray[i];
}

// Calculate slope and intercept
let slope = (count * xySum - xSum * ySum) / (count * xxSum - xSum * xSum);
let intercept = (ySum / count) - (slope * xSum) / count;

// Generate values
const xValues = [];
const yValues = [];
for (let x = 50; x <= 150; x += 1) {
xValues.push(x);
yValues.push(x * slope + intercept);
}

const data = [
{x:xArray, y:yArray, mode:"markers"},
{x:xValues, y:yValues, mode:"line"}
];

const layout = {
xaxis: {range: [40, 160], title: "Square Meters"},
yaxis: {range: [5, 16], title: "Price in Millions"},
title: "House Prices vs. Size"
};

Plotly.newPlot("myPlot", data, layout);
</script>

</body>
</html>

Polynomial Regression

If scattered data points do not fit a linear regression (a straight line through the points), the data may fit an polynomial regression.

A Polynomial Regression, like linear regression, uses the relationship between the variables x and y to find the best way to draw a line through the data points.

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