Simple Linear Regression: What’s inside?

Regression is a statistical approach that suggests predicting a dependent variable (goal feature) with the help of other independent variables (data). Regression is one of the most known and understood statistical methods.

Linear regression is a model that assumes a linear relationship between its dependent and independent variables. Linear regression further branches out to Simple Linear Regression (SLR) and Multiple Linear Regression (MLR). We will explore Single Linear Regression, regression with one dependent and one independent variable, because of its simplicity. SLR’s math is the base of many other machine learning models.

Here I will elaborate on Simple Linear Regression to gain intuition on how it works. I will use an NBA Game Score dataset (link below) to demonstrate SLR and finally compare it to Scikit-learn’s Linear Regression model.

Simple Linear Regression

To understand SLR, let’s break down the concepts we must go through

  • SLR line and its coefficients
  • Loss function
  • Gradient descent
  • Deriving the coefficients (optional)

SLR Line and its coefficients

The slope intercept form of a line is Y= MX+B.

Y is the dependent variable (Goal), X is the independent variable (Data), M and B are the characteristics of the line. Slope (M) gives us how related variables X and Y are and Intercept (B) gives us information on the value of the dependent variable when the rate of change is eliminated.

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In SLR the equation is written as **y = b0 + x b1. **b0 and b1 are the intercept and slope respectively. They are determined by the given formulas below to find the line of best fit.

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But, this is not always the case in regression. Let’s understand why in the gradient descent section. If you are curious about how we stumbled upon these, do check out the optional section.

Loss Function

The loss function is a metric that suggests how much the predicted value deviates from the actual value. There are plenty of loss functions available, we will look at Mean Squared Error (MSE).

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MSE, as the name suggests, squares the difference between the actual and predicted value for each record, sums it up and divides it by the number of records. Our goal is to find a model that yields the smallest loss.

Gradient Descent

Gradient descent is an optimizing algorithm that updates the parameters iteratively to find the model with the slightest loss. The loss function for a model with one or two parameters can be partially differentiated to find the minimum. But as our dimensions increase it’s hard to visualize the parameters, let alone the eigenvalues for each solution. Due to multiple occurrences of local minima, we will have to traverse through all the combinations of eigenvalues to make sure we found the global minima. Although the global minima problem is not fully rectified, Gradient Descent helps to find the minima for models with higher orders.

But we haven’t explored the base of the problem: the loss function. MSE (being a quadratic function) guarantees us there will always be a point on the curve whose gradient is zero, but there are loss functions that do not guarantee a point with zero gradient or the point with a zero gradient might not always be the global minima. To overcome this problem gradient descent is employed.

Specific to our situation, we can choose to find the coefficients by the formulae presented above or we can start with random non zero values and let it work its way to the best fit. The mathematical significance of the gradient descent algorithm deserves an article for itself. For now, I will go through the intuition required to implement the algorithm. The mathematical approach is similar to that of the coefficients, I figured it would be redundant to include it (I will link it in the end if you are curious).

Now, let’s see what gradient descent is all about. Imagine a person is hiking downhill with no optical senses; the person’s goal is to reach the bottom of the valley. Intuitively he takes a step forward, if the slope is downward he would continue to move until he encounters a change in slope. Once the person feels no elevation while moving, he/she stops.

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But as expressed above, it wouldn’t make sense to take steps of a defined length and then evaluate for course correction, as the person could have passed the minimum only to realize he/she moved in the wrong direction. This is where the learning rate comes into the picture. It penalizes large steps to make sure the person does not take a step over the minimum.

#statistics #machine-learning #data-science #deep learning

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Simple Linear Regression: What’s inside?
Mike  Kozey

Mike Kozey

1656151740

Test_cov_console: Flutter Console Coverage Test

Flutter Console Coverage Test

This small dart tools is used to generate Flutter Coverage Test report to console

How to install

Add a line like this to your package's pubspec.yaml (and run an implicit flutter pub get):

dev_dependencies:
  test_cov_console: ^0.2.2

How to run

run the following command to make sure all flutter library is up-to-date

flutter pub get
Running "flutter pub get" in coverage...                            0.5s

run the following command to generate lcov.info on coverage directory

flutter test --coverage
00:02 +1: All tests passed!

run the tool to generate report from lcov.info

flutter pub run test_cov_console
---------------------------------------------|---------|---------|---------|-------------------|
File                                         |% Branch | % Funcs | % Lines | Uncovered Line #s |
---------------------------------------------|---------|---------|---------|-------------------|
lib/src/                                     |         |         |         |                   |
 print_cov.dart                              |  100.00 |  100.00 |   88.37 |...,149,205,206,207|
 print_cov_constants.dart                    |    0.00 |    0.00 |    0.00 |    no unit testing|
lib/                                         |         |         |         |                   |
 test_cov_console.dart                       |    0.00 |    0.00 |    0.00 |    no unit testing|
---------------------------------------------|---------|---------|---------|-------------------|
 All files with unit testing                 |  100.00 |  100.00 |   88.37 |                   |
---------------------------------------------|---------|---------|---------|-------------------|

Optional parameter

If not given a FILE, "coverage/lcov.info" will be used.
-f, --file=<FILE>                      The target lcov.info file to be reported
-e, --exclude=<STRING1,STRING2,...>    A list of contains string for files without unit testing
                                       to be excluded from report
-l, --line                             It will print Lines & Uncovered Lines only
                                       Branch & Functions coverage percentage will not be printed
-i, --ignore                           It will not print any file without unit testing
-m, --multi                            Report from multiple lcov.info files
-c, --csv                              Output to CSV file
-o, --output=<CSV-FILE>                Full path of output CSV file
                                       If not given, "coverage/test_cov_console.csv" will be used
-t, --total                            Print only the total coverage
                                       Note: it will ignore all other option (if any), except -m
-p, --pass=<MINIMUM>                   Print only the whether total coverage is passed MINIMUM value or not
                                       If the value >= MINIMUM, it will print PASSED, otherwise FAILED
                                       Note: it will ignore all other option (if any), except -m
-h, --help                             Show this help

example run the tool with parameters

flutter pub run test_cov_console --file=coverage/lcov.info --exclude=_constants,_mock
---------------------------------------------|---------|---------|---------|-------------------|
File                                         |% Branch | % Funcs | % Lines | Uncovered Line #s |
---------------------------------------------|---------|---------|---------|-------------------|
lib/src/                                     |         |         |         |                   |
 print_cov.dart                              |  100.00 |  100.00 |   88.37 |...,149,205,206,207|
lib/                                         |         |         |         |                   |
 test_cov_console.dart                       |    0.00 |    0.00 |    0.00 |    no unit testing|
---------------------------------------------|---------|---------|---------|-------------------|
 All files with unit testing                 |  100.00 |  100.00 |   88.37 |                   |
---------------------------------------------|---------|---------|---------|-------------------|

report for multiple lcov.info files (-m, --multi)

It support to run for multiple lcov.info files with the followings directory structures:
1. No root module
<root>/<module_a>
<root>/<module_a>/coverage/lcov.info
<root>/<module_a>/lib/src
<root>/<module_b>
<root>/<module_b>/coverage/lcov.info
<root>/<module_b>/lib/src
...
2. With root module
<root>/coverage/lcov.info
<root>/lib/src
<root>/<module_a>
<root>/<module_a>/coverage/lcov.info
<root>/<module_a>/lib/src
<root>/<module_b>
<root>/<module_b>/coverage/lcov.info
<root>/<module_b>/lib/src
...
You must run test_cov_console on <root> dir, and the report would be grouped by module, here is
the sample output for directory structure 'with root module':
flutter pub run test_cov_console --file=coverage/lcov.info --exclude=_constants,_mock --multi
---------------------------------------------|---------|---------|---------|-------------------|
File                                         |% Branch | % Funcs | % Lines | Uncovered Line #s |
---------------------------------------------|---------|---------|---------|-------------------|
lib/src/                                     |         |         |         |                   |
 print_cov.dart                              |  100.00 |  100.00 |   88.37 |...,149,205,206,207|
lib/                                         |         |         |         |                   |
 test_cov_console.dart                       |    0.00 |    0.00 |    0.00 |    no unit testing|
---------------------------------------------|---------|---------|---------|-------------------|
 All files with unit testing                 |  100.00 |  100.00 |   88.37 |                   |
---------------------------------------------|---------|---------|---------|-------------------|
---------------------------------------------|---------|---------|---------|-------------------|
File - module_a -                            |% Branch | % Funcs | % Lines | Uncovered Line #s |
---------------------------------------------|---------|---------|---------|-------------------|
lib/src/                                     |         |         |         |                   |
 print_cov.dart                              |  100.00 |  100.00 |   88.37 |...,149,205,206,207|
lib/                                         |         |         |         |                   |
 test_cov_console.dart                       |    0.00 |    0.00 |    0.00 |    no unit testing|
---------------------------------------------|---------|---------|---------|-------------------|
 All files with unit testing                 |  100.00 |  100.00 |   88.37 |                   |
---------------------------------------------|---------|---------|---------|-------------------|
---------------------------------------------|---------|---------|---------|-------------------|
File - module_b -                            |% Branch | % Funcs | % Lines | Uncovered Line #s |
---------------------------------------------|---------|---------|---------|-------------------|
lib/src/                                     |         |         |         |                   |
 print_cov.dart                              |  100.00 |  100.00 |   88.37 |...,149,205,206,207|
lib/                                         |         |         |         |                   |
 test_cov_console.dart                       |    0.00 |    0.00 |    0.00 |    no unit testing|
---------------------------------------------|---------|---------|---------|-------------------|
 All files with unit testing                 |  100.00 |  100.00 |   88.37 |                   |
---------------------------------------------|---------|---------|---------|-------------------|

Output to CSV file (-c, --csv, -o, --output)

flutter pub run test_cov_console -c --output=coverage/test_coverage.csv

#### sample CSV output file:
File,% Branch,% Funcs,% Lines,Uncovered Line #s
lib/,,,,
test_cov_console.dart,0.00,0.00,0.00,no unit testing
lib/src/,,,,
parser.dart,100.00,100.00,97.22,"97"
parser_constants.dart,100.00,100.00,100.00,""
print_cov.dart,100.00,100.00,82.91,"29,49,51,52,171,174,177,180,183,184,185,186,187,188,279,324,325,387,388,389,390,391,392,393,394,395,398"
print_cov_constants.dart,0.00,0.00,0.00,no unit testing
All files with unit testing,100.00,100.00,86.07,""

Installing

Use this package as an executable

Install it

You can install the package from the command line:

dart pub global activate test_cov_console

Use it

The package has the following executables:

$ test_cov_console

Use this package as a library

Depend on it

Run this command:

With Dart:

 $ dart pub add test_cov_console

With Flutter:

 $ flutter pub add test_cov_console

This will add a line like this to your package's pubspec.yaml (and run an implicit dart pub get):

dependencies:
  test_cov_console: ^0.2.2

Alternatively, your editor might support dart pub get or flutter pub get. Check the docs for your editor to learn more.

Import it

Now in your Dart code, you can use:

import 'package:test_cov_console/test_cov_console.dart';

example/lib/main.dart

import 'package:flutter/material.dart';

void main() {
  runApp(MyApp());
}

class MyApp extends StatelessWidget {
  // This widget is the root of your application.
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      theme: ThemeData(
        // This is the theme of your application.
        //
        // Try running your application with "flutter run". You'll see the
        // application has a blue toolbar. Then, without quitting the app, try
        // changing the primarySwatch below to Colors.green and then invoke
        // "hot reload" (press "r" in the console where you ran "flutter run",
        // or simply save your changes to "hot reload" in a Flutter IDE).
        // Notice that the counter didn't reset back to zero; the application
        // is not restarted.
        primarySwatch: Colors.blue,
        // This makes the visual density adapt to the platform that you run
        // the app on. For desktop platforms, the controls will be smaller and
        // closer together (more dense) than on mobile platforms.
        visualDensity: VisualDensity.adaptivePlatformDensity,
      ),
      home: MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  MyHomePage({Key? key, required this.title}) : super(key: key);

  // This widget is the home page of your application. It is stateful, meaning
  // that it has a State object (defined below) that contains fields that affect
  // how it looks.

  // This class is the configuration for the state. It holds the values (in this
  // case the title) provided by the parent (in this case the App widget) and
  // used by the build method of the State. Fields in a Widget subclass are
  // always marked "final".

  final String title;

  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  int _counter = 0;

  void _incrementCounter() {
    setState(() {
      // This call to setState tells the Flutter framework that something has
      // changed in this State, which causes it to rerun the build method below
      // so that the display can reflect the updated values. If we changed
      // _counter without calling setState(), then the build method would not be
      // called again, and so nothing would appear to happen.
      _counter++;
    });
  }

  @override
  Widget build(BuildContext context) {
    // This method is rerun every time setState is called, for instance as done
    // by the _incrementCounter method above.
    //
    // The Flutter framework has been optimized to make rerunning build methods
    // fast, so that you can just rebuild anything that needs updating rather
    // than having to individually change instances of widgets.
    return Scaffold(
      appBar: AppBar(
        // Here we take the value from the MyHomePage object that was created by
        // the App.build method, and use it to set our appbar title.
        title: Text(widget.title),
      ),
      body: Center(
        // Center is a layout widget. It takes a single child and positions it
        // in the middle of the parent.
        child: Column(
          // Column is also a layout widget. It takes a list of children and
          // arranges them vertically. By default, it sizes itself to fit its
          // children horizontally, and tries to be as tall as its parent.
          //
          // Invoke "debug painting" (press "p" in the console, choose the
          // "Toggle Debug Paint" action from the Flutter Inspector in Android
          // Studio, or the "Toggle Debug Paint" command in Visual Studio Code)
          // to see the wireframe for each widget.
          //
          // Column has various properties to control how it sizes itself and
          // how it positions its children. Here we use mainAxisAlignment to
          // center the children vertically; the main axis here is the vertical
          // axis because Columns are vertical (the cross axis would be
          // horizontal).
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            Text(
              'You have pushed the button this many times:',
            ),
            Text(
              '$_counter',
              style: Theme.of(context).textTheme.headline4,
            ),
          ],
        ),
      ),
      floatingActionButton: FloatingActionButton(
        onPressed: _incrementCounter,
        tooltip: 'Increment',
        child: Icon(Icons.add),
      ), // This trailing comma makes auto-formatting nicer for build methods.
    );
  }
}

Author: DigitalKatalis
Source Code: https://github.com/DigitalKatalis/test_cov_console 
License: BSD-3-Clause license

#flutter #dart #test 

A Deep Dive into Linear Regression

Let’s begin our journey with the truth — machines never learn. What a typical machine learning algorithm does is find a mathematical equation that, when applied to a given set of training data, produces a prediction that is very close to the actual output.

Why is this not learning? Because if you change the training data or environment even slightly, the algorithm will go haywire! Not how learning works in humans. If you learned to play a video game by looking straight at the screen, you would still be a good player if the screen is slightly tilted by someone, which would not be the case in ML algorithms.

However, most of the algorithms are so complex and intimidating that it gives our mere human intelligence the feel of actual learning, effectively hiding the underlying math within. There goes a dictum that if you can implement the algorithm, you know the algorithm. This saying is lost in the dense jungle of libraries and inbuilt modules which programming languages provide, reducing us to regular programmers calling an API and strengthening further this notion of a black box. Our quest will be to unravel the mysteries of this so-called ‘black box’ which magically produces accurate predictions, detects objects, diagnoses diseases and claims to surpass human intelligence one day.

We will start with one of the not-so-complex and easy to visualize algorithm in the ML paradigm — Linear Regression. The article is divided into the following sections:

  1. Need for Linear Regression

  2. Visualizing Linear Regression

  3. Deriving the formula for weight matrix W

  4. Using the formula and performing linear regression on a real world data set

Note: Knowledge on Linear Algebra, a little bit of Calculus and Matrices are a prerequisite to understanding this article

Also, a basic understanding of python, NumPy, and Matplotlib are a must.


1) Need for Linear regression

Regression means predicting a real valued number from a given set of input variables. Eg. Predicting temperature based on month of the year, humidity, altitude above sea level, etc. Linear Regression would therefore mean predicting a real valued number that follows a linear trend. Linear regression is the first line of attack to discover correlations in our data.

Now, the first thing that comes to our mind when we hear the word linear is, a line.

Yes! In linear regression, we try to fit a line that best generalizes all the data points in the data set. By generalizing, we mean we try to fit a line that passes very close to all the data points.

But how do we ensure that this happens? To understand this, let’s visualize a 1-D Linear Regression. This is also called as Simple Linear Regression

#calculus #machine-learning #linear-regression-math #linear-regression #linear-regression-python #python

Angela  Dickens

Angela Dickens

1598352300

Regression: Linear Regression

Machine learning algorithms are not your regular algorithms that we may be used to because they are often described by a combination of some complex statistics and mathematics. Since it is very important to understand the background of any algorithm you want to implement, this could pose a challenge to people with a non-mathematical background as the maths can sap your motivation by slowing you down.

Image for post

In this article, we would be discussing linear and logistic regression and some regression techniques assuming we all have heard or even learnt about the Linear model in Mathematics class at high school. Hopefully, at the end of the article, the concept would be clearer.

**Regression Analysis **is a statistical process for estimating the relationships between the dependent variables (say Y) and one or more independent variables or predictors (X). It explains the changes in the dependent variables with respect to changes in select predictors. Some major uses for regression analysis are in determining the strength of predictors, forecasting an effect, and trend forecasting. It finds the significant relationship between variables and the impact of predictors on dependent variables. In regression, we fit a curve/line (regression/best fit line) to the data points, such that the differences between the distances of data points from the curve/line are minimized.

#regression #machine-learning #beginner #logistic-regression #linear-regression #deep learning

5 Regression algorithms: Explanation & Implementation in Python

Take your current understanding and skills on machine learning algorithms to the next level with this article. What is regression analysis in simple words? How is it applied in practice for real-world problems? And what is the possible snippet of codes in Python you can use for implementation regression algorithms for various objectives? Let’s forget about boring learning stuff and talk about science and the way it works.

#linear-regression-python #linear-regression #multivariate-regression #regression #python-programming

David mr

David mr

1622168430

Implementing Simple Linear Regression in Python from Scratch

Today we will implement one of the most famous machine learning algorithms, Simple Linear Regression from scratch and then with scikit-learn module and then compare the two models. The code for this can be found on this Github Link.

Understanding the Math’s

Implementation :

Conclusion

#linear-regression-python #linear-regression #data-science #python #machine-learning