Thursday, September 22, 2022

How Do You Make Predictions In Math

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New Observations Versus Data Used To Fit The Model

Making predictions with probability | Statistics and probability | 7th grade | Khan Academy

R-squared and S indicate how well the model fits the observed data. We need predictions for new observations that the analysis did not use during the model estimation process. Assessing that type of fit requires a different goodness-of-fit measure, the predicted R-squared.

Predicted R-squared measures how well the model predicts the value of new observations. Statistical software packages calculate it by sequentially removing each observation, fitting the model, and determining how well the model predicts the removed observations.

If the predicted R-squared is much lower than the regular R-squared, you know that your regression model doesnt predict new observations as well as it fits the current dataset. This situation should make you wary of the predictions.

The statistical output below shows that the predicted R-squared is nearly equal to the regular R-squared for our model. We have reason to believe that the model predicts new observations nearly as well as it fits the dataset.

Related post: How to Interpret Adjusted R-squared and Predicted R-squared

Is Perfect Prediction Possible

Big data has made prediction methods increasingly accurate. But can human behavior ever be perfectly predicted?

The most basic equation is that of Y = f, which reads, Y is a function of X. Input a value for X, and the scientist will tell you the likely value for Y. The more complex the model, the more need for more inputs, and so the simple equation gets a whole lot more complicated.

Of course, it doesnt always work out. Hurricanes take trajectories not predicted by weather models. Tumors grow slower or faster than predicted. Scientists, just like anyone else, rarely if ever predict perfectly. No matter what data and mathematical model you have, the future is still uncertain.

So, scientists have to allow for error in our fundamental equation. That is, Y = f + E, where E encompasses our inability to predict perfectly. Its the part of the equation that keeps us humble.

As technology develops, scientists may find that we can predict human behavior rather well in one area, while still lacking in another. Its very difficult to give an overall sense of the limitations. For instance, facial recognition may be easier to emulate because vision is one of many human sensory processing systems, or because there are only so many ways faces can differ. On the other hand, predicting voting behavior, especially based on the 2016 presidential election, is quite another story. There are many complex and not yet understood reasons why humans do what they do.

Why Is Predictive Analytics Important

While organizations have recognized the importance of gathering data as a means of looking back on industry trends for years, business teams have only just started scratching the surface of possibility when it comes to predictive analytics.

Analytics is getting exciting in every industry because were equipped than ever touse the data in the back room that has been gathering dustto make better business decisions, Goulding says.

From insurance to retail to healthcare, organizations are starting to adapt to this model of informed decision-making and are using it to their advantage:

  • Today, insurance companies can predict if a new client is a risk based on their age, history, health conditions, etc. They can weigh this data and make an informed decision about whether or not they want to cover that individual.
  • Retail organizations can predict how new brands or items might sell in their local market based on consumer demographics. They can then make strategic decisions about how much product to stock.
  • Doctors can use predictive data to help determine not only what ailment someones conditions point to but also their chances of survival, whether or not they need immediate surgery, and their conditions expected decline over a certain period of time.

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How Useful Is Mathematics To Humankind

The subject makes a man more methodical and systematic, mathematics makes our life orderly and prevents chaos. Certain qualities that you are nurtured while studying mathematics like creativity, reasoning, abstract or spatial thinking, critical thinking, problem-solving, and even effective communication skills.

Lottery Prediction For The Irish Lotto 6/47

Making predictions

The lottery prediction table above shows the huge difference between Patterns 1 and 210.

If you are an Irish Lottery player, are you willing to waste money on 199 or 210?

The problem, the majority of lottery players never know they are falling into lottery numbers that will never appear in an Irish lottery. I can almost guarantee, you are probably one of these players.

The Irish lottery has 210 total patterns but only three are the best.

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Interpreting The Regression Prediction Results

The output indicates that the mean value associated with a BMI of 18 is estimated to be ~23% body fat. Again, this mean applies to the population of middle school girls. Lets assess the precision using the confidence interval and the prediction interval .

The confidence interval is the range where the mean value for girls with a BMI of 18 is likely to fall. We can be 95% confident that this mean is between 22.1% and 23.9%. However, this confidence interval does not help us evaluate the precision of individual predictions.

A prediction interval is the range where a single new observation is likely to fall. Narrower prediction intervals represent more precise predictions. For an individual middle school girl with a BMI of 18, we can be 95% confident that her body fat percentage is between 16% and 30%.

The range of the prediction interval is always wider than the confidence interval due to the greater uncertainty of predicting an individual value rather than the mean.

Is this prediction sufficiently precise? To make this determination, well need to use our subject-area knowledge in conjunction with any specific requirements we have. Im not a medical expert, but Id guess that the 14 point range of 16-30% is too imprecise to provide meaningful information. If this is true, our regression model is too imprecise to be useful.

Yes I Know Your Math Says X But This Other Math Says Y Which Is Inconsistent With X And Y Seems Much More Intuitively Sound To Me So I Believe Y

No, Y is not inconsistent with X. To someone who understands the math, this is obvious but if you dont understand the math and are relying on natural language descriptions, yes, they certainly can sound inconsistent, particularly since many authors are sloppy in their terminology because they are more concerned with getting across some pictures of what theyre talking about than with strict accuracy and consistency. They dont expect what they write to be taken as a proof of the theory, or a completely consistent explanation of it, just as an attempt to describe some aspect of it in a way that is not going to be judged based on apparent consistency with other aspects.

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Precision Of The Predictions

Previously, we established that our regression model provides unbiased predictions of the observed values. Thats good. However, it doesnt address the precision of those predictions. Precision measures how close the predictions are to the observed values. We want the predictions to be both unbiased and close to the actual values. Predictions are precise when the observed values cluster close to the predicted values.

Regression predictions are for the mean of the dependent variable. If you think of any mean, you know that there is variation around that mean. The same applies to the predicted mean of the dependent variable. In the fitted line plot, the regression line is nicely in the center of the data points. However, there is a spread of data points around the line. We need to quantify that spread to know how close the predictions are to the observed values. If the spread is too large, the predictions wont provide useful information.

Later, Ill generate predictions and show you how to assess the precision.

Related post: Understand Precision in Applied Regression to Avoid Costly Mistakes

What Is Predictive Analytics

7th grade Math Making Predictions

Predictive analytics uses mathematical modeling tools to generate predictions about an unknown fact, characteristic, or event. Its about taking the data that you know exists and building a mathematical model from that data to help you make predictions about somebody not yet in that data set, Goulding explains.

An analysts role in predictive analysis is to assemble and organize the data, identify which type of mathematical model applies to the case at hand, and then draw the necessary conclusions from the results. They are often also tasked with communicating those conclusions to stakeholders effectively and engagingly.

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What Is Mathematics In The Modern World

In a modern world, math such as applied mathematics is not only relevant, its crucial. The idea of applied math is to create a group of methods that solve problems in science. Modern areas of applied math include mathematical physics, mathematical biology, control theory, aerospace engineering, and math finance.

How Can Math Be Used To Predict The Stock Market

Probabilities. No mathematical system, however advanced, can predict the actual future. But sophisticated mathematics can calculate the probability of events. This works in the stock market by helping traders minimize the likelihood that something bad might happen before a certain date or other precursor.

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Create Models And Forecast Future Outcomes

Predictive modeling is a technique that uses mathematical and computational methods to predict an event or outcome. A mathematical approach uses an equation-based model that describes the phenomenon under consideration. The model is used to forecast an outcome at some future state or time based upon changes to the model inputs. The model parameters help explain how model inputs influence the outcome. Examples include time-series regression models for predicting airline traffic volume or predicting fuel efficiency based on a linear regression model of engine speed versus load.

The computational predictive modeling approach differs from the mathematical approach because it relies on models that are not easy to explain in equation form and often require simulation techniques to create a prediction. This approach is often called black box predictive modeling because the model structure does not provide insight into the factors that map model input to outcome. Examples include using neural networks to predict which winery a glass of wine originated from or bagged decision trees for predicting the of a borrower.

Predictive modeling is often performed using curve and surface fitting, time series regression, or machine learning approaches. Regardless of the approach used, the process of creating a predictive model is the same across methods. The steps are:

How Do We Identify The Best Patterns In Lottery

Predictions with Probability

We use math.

Fortunately, the formula to use is the same in any lottery. We simply change the variable depending on the format. So the 5/69 Powerball have different probability analysis compared to 5/59 Eurojackpot or 5/75 Mega Millions.

To apply math in the lottery first, we get the probability of each pattern. Then, we multiply the probability with the number of draws to get its predicted frequency or in simple terms, the estimated occurrence.

Estimated Occurrence = Probability X number of draws

According to The Law Of Large Numbers, the average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer as more trials are performed to infinity.

Therefore, to prove that our recommendation of playing the lottery based on patterns is correct, our estimation should be close to the actual results, given a sufficiently large number of data from actual lottery draws.

Estimated occurrence is equal or close to actual occurrence

So without further ado, lets go to the confirmation part of this article.

Enter actual lottery results.

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The Regression Approach For Predictions

Using regression to make predictions doesnt necessarily involve predicting the future. Instead, you predict the mean of the dependent variable given specific values of the independent variable. For our example, well use one independent variable to predict the dependent variable. I measured both of these variables at the same point in time.

Psychic predictions are things that just pop into mind and are not often verified against reality. Unsurprisingly, predictions in the regression context are more rigorous. We need to collect data for relevant variables, formulate a model, and evaluate how well the model fits the data.

The general procedure for using regression to make good predictions is the following:

  • Research the subject-area so you can build on the work of others. This research helps with the subsequent steps.
  • Collect data for the relevant variables.
  • Specify and assess your regression model.
  • If you have a model that adequately fits the data, use it to make predictions.
  • While this process involves more work than the psychic approach, it provides valuable benefits. With regression, we can evaluate the bias and precision of our predictions:

    • Bias in a statistical model indicates that the predictions are systematically too high or too low.
    • Precision represents how close the predictions are to the observed values.

    How Accurate Are The Predictions

    We assess the accuracy of our predictions using two statistics, standard error and correlation.

    • Standard error is a measure of how close the predictions were to the actual exam outcomes effectively it is the standard deviation of the differences between the two. For example, the standard errors on our GCSE predictions are typically just over a grade for 9-1 GCSEs. The chances graphs themselves provide a useful illustration of what this means in practice.
    • Correlation is a measure of the extent to which one variable increases as another variable increases. In a scatter plot of one variable versus the other, the strength of the correlation influences the pattern seen in the points, as shown in the examples given in Figures 3 to 5.

    Figure 3: Scatterplot of two variables with a correlation of +0.7

    Figure 4: Scatterplot of two variables with a correlation of -0.7

    Figure 5: Scatterplot of two variables with a correlation of zero

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    How Does Math Help Us Make Predictions

    Article Summary: When it comes to the subject of how mathematics is used as the basis for making predictions a common reaction among people is to shake their heads. Isn’t predicting something that fortune tellers and magicians do they ask?

    When it comes to the subject of how mathematics is used as the basis for making predictions a common reaction among people is to shake their heads. Isn’t predicting something that fortune tellers and magicians do they ask? Well, yes, but what stereotypical fortune tellers really do is make guesses that are not based on sound mathematical principles. They simply pull statements out of the air and hope they come to pass. Usually their predictions falter because they are not based on a statistical analysis of data. Predictions without a basis in math is little more than guessing and guessing is not reliable. This is why people who predict the weather, the outcomes of political elections, the decline and fall of a stock are often closer to accuracy than not.

    Of course, there are other areas where predictions come in handy that are of a more serious nature. For example, in the stock market operates brokers will collect data about a particular stock or industry. Then, they will look at the common external factors that can manipulate how the stocks go up or down. Again, the predictions are based on collected data and patterns.

    Predictive Analytics: What It Is & Why Its Important

    Grade 7 Math #12.2b, Make predictions with Simulations for experimental probability

    Over the last decade, organizations across industries have come to rely on the 2.5 quintillion bytes of data humans generate daily to understand their consumers better, identify patterns in behavior, and make more effective and strategic .

    As the technology used to collect and analyze data has continued to advance, these organizations have evolved their data-related practices with it. Data analysts can now achieve a level of insight that extends beyond a description of past behavior and instead use data strategically to look ahead at future possibilities.

    Data analytics today is allowing us for the first time to take the massive amount of data weve been assembling for years and use it for predictive purposes rather than in just descriptive ways, saysThomas Goulding, professor for theMaster of Professional Studies in Analytics program withinNortheasterns College of Professional Studies.Through the use of mathematical modeling and data analytics, data can now tell me something I wouldnt otherwise have been able to learn. As a result, because of analytics, we can make informed business decisions today that simply were not possible ten years ago.

    Known as predictive analytics, this new application of data analysis has successfully served an array of vital industry needs. Read on to explore what predictive analytics entails, examples of its many uses across sectors, and the skills you need to succeed in this ever-changing field.

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    Lottery Prediction For The Uk Lotto 6/59

    The above table shows that patterns 434 and 462 will not appear. If you were to play the UK lottery, you will not want to waste your money with these kinds of patterns. Will you?

    UK Lotto has only one best pattern. There are 30 patterns that are not so good and 431 that are worst.

    Whats the guarantee you are not picking lotto numbers based on these 431 worst combinations?

    Does Mathematicians Have A Future

    The progression of both the nature of mathematics and individual mathematical problems into the future is a widely debated topic many past predictions about modern mathematics have been misplaced or completely false, so there is reason to believe that many predictions today may follow a similar path.

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