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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 AI, a Linear Regression is a supervised machine learning algorithm.

Scatter Plot

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

Example

var xArray = [50,60,70,80,90,100,110,120,130,140,150];
var yArray = [7,8,8,9,9,9,10,11,14,14,15];

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

// Define Layout
var 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);
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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

var xArray = [50,60,70,80,90,100,110,120,130,140,150];
var yArray = [7,8,8,9,9,9,9,10,11,14,14,15];

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

var 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);
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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

var xArray = [50,60,70,80,90,100,110,120,130,140,150];
var yArray = [7,8,8,9,9,9,10,11,14,14,15];

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

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

var xArray = [50,60,70,80,90,100,110,120,130,140,150];
var yArray = [7,8,8,9,9,9,10,11,14,14,15];

// Calculate Sums
var xSum=0, ySum=0 , xxSum=0, xySum=0;
var count = xArray.length;
for (var 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
var slope = (count * xySum - xSum * ySum) / (count * xxSum - xSum * xSum);
var intercept = (ySum / count) - (slope * xSum) / count;

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