Machine Learning Full Course 2026 | Machine Learning Tutorial | Machine Learning | Simplilearn

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[Music] Hey there, welcome to our machine learning full course by Simply Learn. You know how your phone seems to read your mind like when Google finishes typing your sentence before you're done or Netflix somehow picks the perfect show for you Friday night? That's not magic. That's machine learning and you're about to learn how it all works. Here's what's crazy. Machine learning is literally everywhere around us. It's helping doctors spot diseases earlier bank catch fraud and even helping your favorite app show you exactly what you want to see. And the best part is companies are desperately hiring for people who understand this stuff. We are talking about real money here. In India, machine learning experts can earn 10 lakh to 25 lakh perom while in US it can go up to $120,000 plus. And don't worry if you're starting from zero, we have got you covered. We'll begin with simple concept and gradually work up to building actually smart system. You'll learn cool techniques like how computers recognize pattern, make predictions and even detect fake news. By the end of this course, you will be the person who knows how to make computers think and learn. So let's get started. Here's a quick information. If you're interested in launching a high growth career in artificial intelligence and machine learning, this program might be the best thing you have ever come across today. The professional certificate in AI and machine learning offered by Purda University online in collaboration with simply learn and IBM isn't just another course. It's a complete career transforming experience. Ranked one online EI ML certification by career karma. This program is designed to help you master the most in demand skills in AI automation GBT geni LMS deep learning agentic framework and so much more. So whether you're just starting out or looking to upskill, you'll get hands-on with 15 plus durable projects, hugging face, tensorflow, majour, and even build llm based application. So what are you waiting for? Hurry up and enroll now and you can find the course link below. Welcome to math refresher, probability and statistics. In this lesson, we are going to explain the concepts of statistics and probability. Describe conditional probability. Define the chain rule of probability. Discuss the measure of variance. Identify the types of gshian distribution. Basic of statistics and probability. Probability and statistics. Data science relies heavily on estimates and predictions. A significant portion of data science is made up of evaluations and forecast. Statistical methods are used to make estimates for further analysis. Probability theory is helpful for making predictions. Statistical methods are highly dependent on probability theory and all probability and statistics are dependent on data. Data is information acquired for reference or research via observations, facts, and measurements. Data is a set of facts structured in the form that computers can interpret such as numbers, words, estimations, and views. Importance of data. Data aids in seeing more about the information by identifying possible connections between two features. Data assists in the detection of distortion by uncovering hidden patterns based on prior information patterns. Data may be utilized to anticipate the future or predict the current state of affairs. Also, data aids in determining whether two pieces of information have any instance in common or not. Types of data. Data might be quantitative that is data that can be measured or counted in numbers or it may be qualitative which is data which is generally divided into groups or in simpler words which cannot be counted or measured in numbers. Let's consider an example. A customer information data of a bank may contain quantitative and qualitative data. Consider this snapshot where we have customer ID, surname, geography, gender, age, balance, has C or card is active member. Amongst these variables we can see surname is mostly qualitative as it cannot be counted and measured in numbers. Geography and gender are also qualitative as they cannot be counted in numbers and are mostly groups. has C or card that is has credit card and is active member although are containing numerical in form but these are categorical that means these have been divided into groups of one and zero that represent yes and no as an answer hence these two variables are also qualitative customer ID is again although a numerical data however the significance or intuition behind Customer ID is categorical. Hence, it may be kept in the qualitative data also. However, age and balance these are numerical information which have been measured or counted and numerical operations can be performed on them. Hence, these are under quantitative data categories. Introduction to descriptive statistics. Descriptive statistics. A descriptive measurement is summary measure that quantitatively portrays the most important features of a set of data allowing for a better comprehension of the information. Data can be measured as different levels. The levels of measurement describe the nature of information stored in the data assigned to the variables. Qualitative data can be measured as nominal or ordinal. Quantitative data can be measured in terms of interval and ratio type. Nominal data. The data is categorized using names, labels or qualities. For example, brand name, zip code, and gender. Ordinal data can be arranged in order or ranked and can be compared. Examples include grades, star reviews, position, and race, and date. Interval data is the data that is ordered and has meaningful differences between the data points. Example, temperature in Celsius and year of birth. Ratio data is similar to the interval level with the added property of inherent zero. Mathematical calculations can be performed on both interval as well as ratio data. For example, height, age, and weight. Population versus sample. Before analyzing the data, it's important to figure out if it's from a population or a sample. Population is a collection of all available items as well as each unit in our study. Sample is a subset of the population that contains only a few units of the population. Population data is used for study when the data pool is very small and can give all the required information. Samples are collected randomly and represent the entire population in the best possible way. Measures of central tendency. The central tendency is a single value that aids in the description of the data by determining its center position. Measures of central tendency are sometimes known as summary statistics or measures of central location. The most popular measurements of central tendency are mean, median, and mode. The normal distribution is a bell-shaped symmetrical distribution in which mean, median, and mode all are equal. The curve over here shows the bell-shaped curve or the normal distribution of variable X. The point over here that is X1 is the point which represents the mean, median and mode of this distribution. Mean mean is calculated by dividing these sum of all data values by the total number of data values. It gets affected when there are unusual or extreme values. It is sensitive to the outliers. Mean can be calculated as summation over all the values of X in a collection divided by the size of the collection. For example, we have a collection where we have values as 7 3 4 1 6 and 7. We find out the sum of these values which is 28 and there are total of six values. So 28 / 6 gives us a mean value of 4.66. Median, it is the middle value in the set of the data that has been sorted in ascending order. It is a better alternative to mean since it is less impacted by outliers and skewess. It is closer to the actual central value. Median is calculated differently for different sizes of data. Differentiated as if the total number of values is odd or if the total number of values is even. If the size of the data is odd. For example, in this case we have five elements. After sorting whatever middle value we get that means n + 1 by 2 term in this case 5 + 1 / 2 that is the third term which is 4 is the median value. In case when the total number of values is even like here there are six values the average or the mean of the two central values is considered as the median. In this case the median is the mean of six and four which is five. Mode. Mode represents the most common value in the data set. It is not at all affected by extreme observations. It is the best measure of central tendency for highly skewed or non-normal distribution. Mode for categorical data is determined by estimating the frequencies for each categories and then the category with the highest frequency is considered to be mode. Like in this case seven has the highest frequency. Hence seven becomes the mode value. However, in case of continuous data or quantitative data, the calculation of mode is slightly different. The first step in calculation of mode is dividing the data into classes which are equal with then getting the frequency of data points lying in within that range of classes and finally selecting the class with the highest frequency. Using the range of that class and the frequencies, we can get the final mode value. Using the formula L plus FM minus F_sub_1 multiplied to H / FM minus F_sub_1 plus FM minus F_sub_2. Here L is the lower limit or the lower observation of the mode class. H is the size of the mode class. FM is the frequency of the mode class. F_sub_1 is the frequency of the class proceeding to mode and F_sub_2 is the frequency of the class succeeding to mode. This gives us the final mode value. Mean versus expectation. Now let's talk about mean versus expectation. So in general we use the expected value or expectation when we want to calculate the mean of a probability distribution that represents the average value we expect to occur before collecting any data. And mean on the other hand mean is basically used when we want to calculate the average value of a given sample. This represents the average value of raw data that we may have already collected. We can understand this by using a simple example. Now to calculate the expected value of this probability distribution, we can use a specific formula from the previous discussion. This is going to be the expected value where X is going to be the data value and this PX is the probability of value. For example, we could calculate the expected value for this probability distribution to be as shown. So here it will be 1.45 goals. So this represents the expected number of goals that the team will score in any given game. And then if you talk about calculating mean, so we typically calculate the mean after we have actually collected raw data. For example, suppose we record the number of goals that a soccer team will score in 15 different games. Now to calculate the mean number of goals scored per game, we can use the following formula where sum of x is basically the sum of all the goals divided by n and the number of records or we can say the sample size. It is as shown on the screen. So this represents the mean number of goals scored per game by the team. Measures of asymmetry. The difference between the three distinct curves can be studied in this image. The central curve is the normal or no skewess curve. Here mean, median and mode all lie on the same point. This normal curve is symmetrical about its mean, median and mode. That means the left hand side of the curve is a mirror image of the right hand side of the curve. However, in case of negatively skewed data, the tail is elongated on the left hand side and the mean is smaller than the mode and the median values or is on the left hand side of the mode. Hence indicating that the outliers are in the negative direction. On the other hand, in case of positively skewed, the data is concentrated on the left hand side of the curve. While the tail is elongated or longer on the right hand side of the curve, the mean is greater than the mode and median or is on the right hand side of the mode and median indicating that the outliers are in the positive direction. Let's consider an example. The graph here shows the global income distribution for the year 2003 2013 and a projection for 2035. If we see the global income distribution statistics for 2003 it is highly right skewed. We can observe in the previous graph that in 2003 the mean of $3,451 was higher than the median of $1090. The global income is definitely not evenly distributed. The majority of people make less than $2,000 each year, while only a small percentage of the population earns more than $14,000. Measures of variability. Measures of variability. Dispersion. The measure of central tendencies provide a single value that addresses the full worth. However, the central tendency cannot depict the viewpoint entirely. The metric of dispersion helps us focus on the inconsistency in the data spread. Measures of dispersion describe the spread of the data. The range, intercortile range, standard deviation and variance are examples of dispersion measures. Range. The range of distribution is the difference between the largest and the smallest amount of data. The range, for example, does not include all of a series positive aspects. It concentrates on the most shocking aspects and ignores that aren't considered critical. For example, for a set 13, 33, 45, 67, 70, the range is 57. That is the maximum of this which is 70 minus the minimum over here which is 13. Variance. Variance is the average of all squared deviations. It is defined as the sum of squared distance between each point and the mean or the dispersion around the mean. The standard deviation is used as variance suffers from a unit difference. Variance can be computed as sigma square summation over x - mu^ 2 divided by n where mu is the mean of the data, x is the individual data point and n is the size of the data. This representation is for a population data. for a sample data variance can be computed as x minus xar whole square summation over it divided by n minus one. Here xbar is the mean of these sample data and n is the sample size. The units of values and variance are not equal. So another variability measure is used. Standard deviation. Standard deviation is a statistical term used to measure the amount of variability or dispersion around a mean. The standard deviation is calculated as the square root of variance. It depicts the concentration of the data around the mean of the data set. Standard deviation as indicated previously can be computed as square root of variance for a population data. Standard deviation sigma can be computed as square root of summation over x i minus mu^ square / n where mu is the mean of the data x i are the data points and n is the size. Let's consider an example. Let's find out the mean, variance, and standard deviation for this data. The data values are three, 5, 6, 9, and 10. To find out the mean, we first find the sum of all these data values that is 33 and divide it by the count, which is five. We get the mean of 6.6. To compute the variance, we start by computing the deviation that is x minus the mean of x. Here 3 is one of the values of the data and 6.6 is the mean. So 3 - 6.6 squared and we do that to find out sum of all the deviations divided by the count which is five. We end up getting an overall variance of 6.64. Standard deviation as we know is measured at square root of variance that is square<unk> of 6.64 which amounts to 2.576. Measures of relationship. Measures of relationship coariance. Coariance is the measure of joint variability of two variables. It measures the direction of the relationship between the variables. It determines if one variable will cause the other to alter in the same way. Coariance between variable X and Y can be computed as summation over the product of X I - XR and Y I - Y bar the whole divided by N minus one. Here Xar and Y bar are the mean of X and Y respectively. The value of covariance can range from minus infinity to a plus infinity. Correlation. Correlation is normalized coariance. It measures the strength of association between two variables. The most common measure for correlation is the Pearson correlation coefficient. Correlation between two variables X and Y can be measured with respect to coariance as coariance between X and Y divided by the standard deviation of X and standard deviation of Y. The value of correlation ranges from a negative 1 to positive 1. Types of correlation. Correlation can be either a positive correlation, zero correlation or a negative correlation. The first picture over here represents a perfect positive correlation wherein a straight line with a positive slope is representing the relationship between the two variables. Zero correlation means that the line representing the relationship between the two variables is horizontal to the xaxis. Perfect negative correlation can be represented by a straight line with a negative slope. Correlation equals to 1 implies a positive relationship. That is when one variable increases the other variable also increases. A correlation value of negative 1 implies a negative relationship. That is when one variable increases the other decreases. The correlation coefficient of zero shows that the variables are completely independent of each other. Let's consider an example. Here we have two variables height and weight. To compute the correlation between height and weight, we use the correlation formula as coariance of X and Y divided by standard deviation of X and standard deviation of Y. Here height is the X variable and weight is the Y variable. First to compute coariance we compute the x - xar and y - y bar values and then the product of them. We then compute x - xr² and y - y bar square values to compute the standard deviations of height and weight respectively. Correlation as we know has been defined as covariance of x and i and y divided by standard deviations of x and y. This can also be represented as summation over x - xr multiplied to y - y bar divided by square root of summation over sum of squared deviations that is x - xr square multiplied to square root of summation over y - y bar square that is sum of square deviations for y. Now let's find out values to put into this formula. First we find out the overall sum of height to get the mean of height which is 5.14. Similarly we get the sum of weight to get the mean of weight as 50. We now get the summation over x - xr multiplied to y - y bar to get the numerator for the formula. Then we compute x - xr square summation and y - y bar square that is sum of squared deviation of x and y respectively. Now we put in the values in this final correlation formula to get a correlation value of 0.889. This indicates that height and weight have a positive relationship. It is evident that as height grows, weight also increases. In this module, we will be talking about expectation and variance. So the expected value or we can say mean of a given variable that we can denote by X is a discrete random variable where it is a weighted average of the possible values that X can take and each value is going to be according to the probability of that specific event occurring. So usually the expected value of X is denoted by a simple formula where we can define the expectation based on the X parameter. which is going to be the sum of each possible outcome multiplied by the probability of the outcome occurring. So in more concrete terms, the expectation is what we would expect the outcome of an experiment to be on average. We can take an example for the coin. If a coin is being tossed 10 times, then one is most likely to get five heads and five tails. Same logic can be discussed if we talk about another example of rolling a die. So there are six possible outcomes when you roll a dieice 1 2 3 4 5 6 and each of these has a probability of 1 by 6 of occurring. So we can say that the expectation is going to be 1 multiplied by the probability of that happening which is going to be 1x 6 + 2x 6 + 3x 6 + 4x 6 + 5x 6 + 6x 6 and that is going to give us 3.5 as an output. The expected value is 3.5. So if you think about it, 3.5 is halfway between the possible values that I can take and this is what we should have expected. Next we talk about the concept of variance. So variance of a random variable allows us to know something about the spread of the possible values of the variable. So for a discrete random variable X the variances of X is going to be denoted by using a simple formula that is going to be var equals E X - M the whole square where M is basically the expected value of the expectation of X. So this is more like a standard deviation of X which can also be represented by using this formula. So the variance does not behave in the same way as expectation when we multiply and add constants to random variables. So now there are two different type of variance that we can have a fair understanding on. First of all we have low variance and then we have high variance. So low variance simply means that there is a small variation in the production of the target function with changes in the trading data set and at the same time high variance as we can see here high variance shows a large variation in prediction of the target function with changes in the trading data set. So a model that shows high variance learns a lot and perform well with the training data set and it does not generalize well with the unseen data set and that's why as a result such a model gives good results with training data set but shows high error rates on the test data set and since the high variance a model learns too much from the data set it leads to an overfitting of the model. So model with high variance will be having couple of issues like it may lead to overfitting or it may also lead to increase in model complexities. Next we have skewess. So skewess in simple terms is basically a measure of asymmetry of a distribution. So distribution is asymmetrical when its left and right sides are not the mirror images. Right now this is a mirrored image and a distribution can have right positive or we can say negative or it can have zero skewess. So right skewed in this scenario is basically the distribution is longer on the right side of its peak and a left skew distribution is going to be we can say where it is longer on the left side. So we can see we have this one as a part of right side. it is more elongated towards the right side and this one is more elongated towards the left side. So we can think of skewess in terms of tails. A tail is long tampering and the end of a distribution. So it simply indicates that they are observations at one end of the distribution but that they are relatively infrequent. So a right skew distribution has a long tail on the right side as you can see here. So the number supports observed. Let's say we have a data on a per year basis. So again we can have a more skewess towards the right side where data is being dropping as we continue to increase the number of years. For example we may have a high sales towards the beginning of year suppose in 2022 but again as we proceed to 2023 second half we are seeing the dip in performance. So that is rightly skewed and same way let's suppose if we started with the sales figure it was really less in suppose 2002 but again as we proceeded to 2023 now our sales have been gradually increasing so it's more like skew towards the left section as a part of negative skew. Next we have curtosis. So curtosis is basically a measure of the tailness of a distribution. So tailness is how often the outliers occur and acts as courtesis is the tailness of the distribution related to a normal distribution. So a distribution with medium curtsis is called as meocurtic. A distribution with low curtosis like this one. This is called as the platicurtic and then distribution with high curtosis like this one. This is called as the leptocortic. So tails here they are tapering ends on either side of a distribution like this. So they represent the probability or the frequency of values that are extremely high or extremely low to the mean. In other words, tails here represents how often the outliers occur. So there are three type of curtsis. We have platocurtic which is negative, leptoccuric which is a positive towards the upper end and then we have messertic which is a normal distribution. So messertic is the medium tail. So normal distributions they have a curtosis of three. So any distribution with a curtsis of a prox value of three is going to be messertic. And curtosis is described in terms of excess curtises which is curtosis minus3. And since normal distribution they have a curtosis of three axis curtises makes comparing a distribution curtosis to a normal distribution even easier. Introduction to probability. Probability theory. Probability is a measure of the likelihood that an event will occur. Let's consider an example of coin toss where the chances of getting heads on a coin are 1 by two or 50%. The probability of each given event is between zero and one both inclusive. Sum of an events cumulative probability cannot be greater than one. Hence the probability of an event X lies between zero and one. This means that the integral of probability of distribution over X equals to 1. Conditional probability. Conditional probability of any event A is defined as the probability of occurrence of A given that event B has previously occurred. Condition probability of event A given B can be estimated as probability of A intersection B that is probability of both A and B happening together divided by the probability of B. It is also written as that probability of A intersection B equals to probability of A given B multiplied to probability of B. Let's consider an example. In a coin, we are doing a two coin flip. Coin one gets heads, tails, heads, and tails in subsequent flips. while coin two gets tails, heads, heads, and tails in the subsequent flips. Now, the probability that coin one will get a head is 2 out of four. While the probability that coin two will get heads is again two out of four. The probability that both coin one and coin two will have a heads is just one out of the four flips. Hence the probability that coin one will get heads given that coin 2 is already heads can be computed as probability of coin one edge intersection coin 2 edge that is 1x4 divided by probability of coin 2 edge that's a given that is 2x 4 which is going to be 0.5 or 50% based base theorem Base theorem calculates the conditional probability of an event based on its prior probabilities. Basically base theorem incorporates the prior probability distribution to predict the posterior probabilities. Base theorem for conditional probability can be expressed as probability of A given B equals probability of B given A divided by probability of B multiplied to probability of A. Base theorem allows updating the probability values by using new information or evidence. Here probability of A is known as prior probability. That is the probability of event before any new data is collected. Probability of A given B is known as the posterior probability. It is the revised probability of an event occurring after taking into consideration the new information probability of B given A is known as the likelihood and probability of B is probability of observing an evidence B model. An example consider an example for calculating the likelihood of having diabetes based on frequency of fast food consumption. Here is the observed data. Let's say the fast food audience is 20%. Diabetes prevalence is 10% and 5% is fast food and diabetes. The chances of diabetes given fast food that is the conditional probability of D given B can be calculated as probability of diabetes and fast food together divided by probability of fast food. That means 5% divided by 20%. that equals 25%. Define an analysis can state eating fast food increases the chance of having diabetes by 25%. The multiplication rule of probability if events A and B are statistically independent and probability of A intersection B can be given as probability of A given B multiplied to probability of B. However, probability of A intersection B is also given as probability of A multiplied to probability of B. Here probability of A given B equals to probability of A when we assume that probability of B is non zero. Similarly, probability of B equals probability of B given A assuming probability of A is non zero. Chain rule of probability joint probability distributions over many random variables can be reduced into conditional distributions over a single variable. It can be expressed as probability of X1 X2 so on until Xn equals probability of X1 intersection probability of X I given probability of X1 till X I minus one. For example, the joint probability of A, B and C can be given as probability of A given B. C multiplied to probability of B given C multiply to probability of C. Logistic sigmoid. The logistics function is a type of sigmoid function that aims to predict the class to which a particular sample belongs. Its outcome is discrete binary value. a probability between zero and one. The logistic sigmoid is a useful function that follows the yes curve. It saturates when the input is very large or very small. Logistic sigmoid is expressed as sigma of x= 1 upon 1 + e to the power minus x. The logistic sigmoid can be expressed as sigmoid function of x is given as 1 upon 1 + e ^ minus x where e is the ooler's number. Gshian distribution. The gossian distribution is a type of distribution in which data tends to cluster around a central value with little or no bias to the left or right. It is often referred to as normal distribution. In absence of prior information, the normal distribution is frequently a fair assumption in machine learning equation. The formula for calculating Gaussian distribution is described as the normal distribution of X. That is the function of X given mean as mu and variance is sigma square can be calculated as 1 upon sigma square<unk> of 2<unk>i. E to the power minus/ X - mood divided by sigma square where mu is the mean or peak value which also is the expected value of X. Sigma is the standard deviation. Sigma square is the variance. A standard normal distribution has a mean of zero and a standard deviation of one. Gshian distribution can be univariate which describes the distribution of a single variable X. It can also be multivariate where it can just use to describe the distribution of several variables. It is represented in 3D of ND formats. Law of large numbers. Now let's talk about law of large numbers. The law of large numbers states that an observed sample average from a large sample will be close to the true population average and that it will get closer in the larger sample. So the law of large number does not guarantee that a given sample spatially a small sample will reflect the true population characteristics or that a sample does not reflect the true population will be balanced by a subsequent sample. This is for the law of large numbers to express the relationship between scale and growth rate. So there are multiple examples through which we can understand and it is widely used in statistical analysis in working with the central limit theorem in terms of the business growth. So there are multiple real time setup in which these are going to be used. So if you talk about tossing a coin, so tossing a coin in a number of times will give us two different type of outcomes. The result will spread evenly between head and tails and the expected average value is going to be half. That means 50 * tails and 30 * heads. But again, if you toss a coin 1,00 times, then the result can be in different manners because out of 1,00 let's say 850 times it has been head and only 150 times it has been tails and so on. So that's why the possibility of one event occurring is going to be changed in large sample sets as compared to a small sample sets as in let's say 10 times. So the number of heads and tails unbalanced for lower number of trials. So we can see it is unbalanced. But again as soon as we toss more number of coins more leans towards the balance value or we can see the observed averages. Next we have P value. So p value is basically a number calculated from the statistical test that describes how likely we are to have found a particular set of observations if the null hypothesis were true. So p values are used in hypothesis testing to help decide whether to reject the null hypothesis. And the smaller the p value, the more likely we are to reject the null hypothesis. So we have a term called as null hypothesis. So all statistical tests they have null hypothesis. So for most tests the null hypothesis is that there is no relationship between our variables of in first or that there is no difference among groups. For example in a two-tail t test the non-hypothesis is that the difference between two groups is going to be zero. So p value is going to tell us how likely it is that our data could have occurred under the null hypothesis. It is done by calculating the likelihood of a test statistic which is the number calculated by a statistical test using our data. So p value tell us how often we would expect to see a test statistic as extreme or more extreme than one calculated by a statistical test. if the null hypothesis of the test was true. So there are multiple limitations as well. So first one is the results can be significant but again they are they may not be practical as we have compared it can be based on multiple hypothesis for a game for the healthcare test. If the test is going to be positive or not it may show even values of the effect of a variable but not the magnitude in real life. What exactly is going to be the application of a drug test being failed in pharma company? Therefore, it is recommended to use confidence and levels in addition to the p values to quantify or we can say to give a solid figure to the reserve which we are going to get. The p values they are interpreted as supporting or we can say refuting the alternative hypothesis. So p value can only tell you whether or not the null hypothesis is supported. It cannot tell us whether our alternative hypothesis is true or why. So the risk of rejecting the null hypothesis is often higher than the p value. So especially when we are looking at a single study or when using small sample sizes. So this is because the smaller frame of reference, the greater are the chance that as we stumble across a statistically significant pattern completely by accident. Key takeaways. Key takeaways. Probability and statistics structure the premise of the data. The data helps in anticipating the future or gauging in view of the past patterns of information. The central tendency is a single value that helps to describe the data by identifying these central positions. The mean, median and mode are the measures of central tendencies. The distribution where the data tends to be around a central value with a lack of bias or minimal bias towards the left or right is called as gshian distribution. >> Mathematics for machine learning. My name is Richard Kersner with the SimplyLearn team. That's get certified, get ahead. We're going to cover mathematics for machine learning. So today's agenda is going to cover data and its types. Then we're going to dive into linear algebra and its concepts, calculus, statistics for machine learning, probability for machine learning, hands-on demos, and of course throwing in there in the middle is going to be your matrixes and a few other things to go along with all this data. Then is types data denotes the individual pieces of factual information collected from various sources. It is stored, processed and later used for analysis. And so we see here uh just a huge grouping of information, a lot of tech stuff, money, dollar signs, numbers uh and then you have your performing analytics to drive insights and hopefully you have a nice share your shareholders gathered at the meeting and you're able to explain it in something they can understand. So we talk about datas types of data we have in our types of data we have a qualitative categorical you think nominal or ordinal and then you have your quantitative or numerical which is discrete or continuous and let's look a little closer at those data type vocabulary always people's favorite is the vocabulary words okay not mine uh but let's dive into this what we mean by nominal nominal they are used to label various just uh label our variables without providing any measurable value. Uh country, gender, race, hair, color, etc. It's something that you either mark true or false. This is a label. It's on or off. Either they have a red hat on or they do not. Uh so a lot of times when you're thinking nominal data labels, uh think of it as a true false kind of setup. And we look at ordinal. This is categorical data with a set order or a scale to it. Uh and you can think of salary range is a great one. Uh movie ratings etc. You see here the salary range if you have 10,000 to 20,000 number of employees earning that rate is 150. 20,000 to 30,000 100 and so forth. Some of the terms you'll hear is bucket. Uh this is where you have 10 different buckets and you want to separate it into something that makes sense into those 10 buckets. And so when we start talking about ordinal, a lot of times when you get down to the brass bones, again, we're talking true false. Uh so if you're a member of the 10 to 20k range, uh so forth, those would each be either part of that group or you're not. But now we're talking about buckets and we want to count how many people are in that bucket. Quantitative numerical data uh falls into two classes, discrete or continuous. And so data with a final set of values which can be categorized class strength questions answered correctly and runs hit in cricket. A lot of times when you see this you can think integer uh and a very restricted integer i.e. you can only have 100 questions um on a test. So you can it's very discreet. I only have a 100 different values that it can attain. So think usually you're talking about integers but within a very small range. They don't have an open end or anything like that. Uh so discrete is very solid, simple to count, set number. Continuous on the other hand uh continuous data can take any numerical value within a range. So water pressure, weight of a person etc. Usually we start thinking about float values where they can get phenomenally small in their in what they're worth. And there's a whole series of values that falls right between discrete and continuous. Um you can think of the stock market. You have dollar amounts. It's still discreet, but it starts to get complicated enough when you have like, you know, jump in the stock market from $525.33 to $580.67. There's a lot of point values in there. It'd still be called discreet, but you start looking at it as almost continuous because it does have such a variance in it. Now uh we talk about n we did we went over nominal and ordinal uh almost true false charts and we looked at quantitative and numerical data which we're starting to get into numbers. Discrete you can usually a lot of times discreet will be put into it could be put into true false but usually it's not. Uh so we want to address this stuff and the first thing we want to look at is the very basic which is your algebra. So we're going to take a look at linear algebra. You can remember back when your uklidian geometry. Uh we have a line. Well, let's go through this. We have linear algebra is the domain of mathematics concerning linear equations and their representations in vector spaces and through matrices. I told you we're going to talk about matrices. Uh so a linear equation is simply um uh 2x + 4 y - 3 z = 10. Very linear. 10 x + 12.4 4 y = z. And now you can actually solve these two equations by combining them. Uh, and that's where we're talking about a linear equation. In the vectors, we have a + b= c. Now, we're starting to look at a direction. And these values usually think of an xyz plot. Um, so each one is a direction. And the actual distance of like a triangle A is C. And then your matrix can describe all kinds of things. Um, I find matrixes uh confuse a lot of people, not because they're particularly difficult, but because of the magnitude and the different things are used for. And a matrix is a chart or a um, you know, think of a spreadsheet, but you have your rows and your columns. And you'll see here we have a * b= c. Very important to know your counts. Uh, so depending on how the math is being done, what you're using it for, making sure you have the same rows and number of columns or a single number, there's all kinds of things that play in that that can make matrixes confusing. Uh, but really it has a lot more to do with what domain you're working in. Uh, are you adding in multiple polomials where you have like uh uh ax^2 plus b y plus, you know, you start to see that can be very confusing versus a very straightforward matrix. And let's just go a little deeper into these because these are such primary this is what we're here to talk about is these different math uh mathematical computations that come up. So we're looking at linear equations. Let's dig deeper into that one. An equation having a maximum order of one is called a linear equation. Uh so it's linear because when you look at this we have uh ax plus b= c which is a one variable. We have two variable ax plus b y = c ax plus b y + z c cz z= d and so forth. But all of these are to the power of one. You don't see x squ. You don't see x cubed. So we're talking about linear equations. That's what we're talking about. In their addition, if you have already dived into say neural networks, you should recognize this ax plus b y plus cz um setup plus the intercept uh which is basically your your neural network each node adding up all the different inputs. And we can drill down into that. Most common formula is your y = mx + c. So you have your uh y equals the m which is your slope, your x value plus c which is your um y intercept. They kind of labeled it wrong here. Threw me for a loop. But the the c would be your y intercept. So when you set x equal to 0, y equals c. And that's that's your y intercept right there. Uh and that's they they just had reversed value of y. When x equals 0, it equals the y intercept, which is c. and your slow gradient line which is your m. So you get your y = 2x + 3. And there's lots of easy ways to compute this. This why this is why we always start with the most basic one when we're solving one of these problems. And then of course the one of the most important takeaways is the slope gradient of the line. Uh so the slope is very important that m value. Uh in this case we went ahead and solved this. If you have y = 2x + 3 you can see how it has a nice line graph here on the right. So matrixes a matrix refers to a rectangular representation of an array of numbers arranged in columns and rows. So we're talking m rows by n columns here a1 is denotes the element of the first row in the first column. Similarly a12 and it's really pronounced a11 in this particular setup. So it's row one column one. A12 is a row one column 2. uh first row and second column and so on. And there's a lot of ways to denote this. I've seen these as like a capital letter a smaller case a for the top row or I mean you can see where they can go all kinds of different directions as far as the value. You just take a moment to realize there's need to be some designation as far as what row it's in and what column it's in. And we have our uh basic operations. We have addition. So when you think about addition, you have uh uh two matrices of 2x two and you just add each individual number in that matrix and then when you get to the bottom you have uh in this case the solution is 12 10 + 2 is 12 5 + 3 is 8 and so on. And the same thing with subtraction. Now again you're counting matrices you want to check your um dimensions of the matrix. the shape. You'll see shape come up a lot in programming. So, we're talking about dimensions. We're talking about the shape. If the two shapes are equal, this is what happens when you add them together or subtract them. And we have multiplication. When you look at the multiplication, you end up with a very uh slightly different setup going. Now, if we look at our last one, we're um uh we're like, why? This always gets to me when we get to matrices because they don't really say why you multiply matrices. Um you know, my first thought is 1 * 2, 4 * 3. But if you look at this, we get 1 * 2 + 4 * 3, 1 * 3 + 4 * 5, uh 6 * 2 + 3 * 3, 6 * 3 + 3 * 5. If you're looking at these matrices, uh, think of this more as an equation. And so we have, uh, if you remember when we back up here for our multiple line equations, let's just go back up a couple slides where we were looking at, uh, two variable. So this is a two variable equation. ax plus b y= c. Um, and this is a way to make it very quick to solve these variables. And that's why you have the matrix, and that's why you do the multiplication the way they do. And this is the dotproduct of uh 1* 2 + 4 * 3 1 * 3 + 4 * 5 uh 6 * 2 + 3 * 3 6 * 3 + 3 * 5 and it gives us a nice little 14 23 21 and 33 over here which then can be used and reduced down to a sample um formula as far as solving the variables as you have enough inputs. Uh and then in matrix operations when you're dealing with a lot of matrices. Uh now keep in mind multiplying matrices is different than finding the product of two matrices. Okay? So when we're talking about multiplication, we're talking about solving uh for equations. When you're finding the product, you are just finding 1* 2. Keep that in mind because that does come up. I've had that come up a number of times where I am altering data and I get confused as to what I'm doing with it. uh transpose flipping the matrix over it's diagonal comes up all the time where you have you still have 12 but instead of it being uh 128 it's now 124 821 you're just flipping the columns and the rows. Uh and then of course you can do an inverse um changing the signs of the values across this main diagonal. And you can see here we have the inverse a to the minus one and ends up with uh instead of 12 8 14 12 it's now -22 -2 vectors uh vector just means we have a value and a direction and we have down four numbers here on our vector. uh in mathematics a one-dimensional matrix is called a vector. Uh so if you have your x plot and you have a single value that values along the x- axis and it's a single dimension. If you have two dimensions you can think about putting them on a graph. You might have x and you might have y and each value denotes a direction. And then of course the actual distance is going to be the hypothesis of that triangle. Uh and you can do that with three dimensionals x y and z. uh and you can do it all the way to nth dimensions. So when they talk about the k means uh for categorizing and how close data is together they will compute that based on the pyagorean theorem. So you would take uh the square of each value, add them all together and find the square root. And that gives you a distance as far as where that point is, where that vector exists or an actual point value. And then you can compare that point value to another one. And makes a very easy comparison versus comparing uh 50 or 60 different numbers. And that brings us up to gene vectors and I gene values. uh igene vectors the vectors that don't change their span while transformation and I gene values the scalar values that are associated to the vectors conceptually you can think of the vector as your picture you have a picture it's um uh two dimensions x and y and so when you do those two dimensions and those two values or whatever that value is um that is that point but the values change when you skew it. And so if we take and we have a vector A and that's a set value. Uh B is um your is your you have A and B which is your hygiene vector. Two is the hyene value. So we're altering all the values by two. That means we're um maybe we're stretching it out one direction, making it tall. Uh if you're doing picture editing, um that one of the places this comes in. But you can see when you're transforming uh your different information, how you transform it is then your hygiene value. And you can see here uh vector after line transition uh we have 3 a. A is the hygiene vector. Three is the hygiene value. So a doesn't change. That's whatever we started with. That's your original picture. And three uh is skewing it one direction and maybe uh b is being skewed another direction. And so you have a nice tilted picture because you've altered it by those by the hygiene values. So let's go ahead and pull up a demo on linear algebra. And to do this, I'm going to go through my trusted Anaconda into my Jupiter notebook. And we'll create a new uh notebook called linear algebra. Since we are working in Python, uh we're going to use our numpy. I always import that as np or numpy array. probably the most popular um module for doing matrixes and things in given that this is part of a series. I'm not going to go too much into numpy. Uh we are going to go ahead and create two different variables. A for a numpy array 105 and b 29. We'll go ahead and run this. And you can see there's our two arrays 105 29. And I went ahead and added a space there in between so it's easier to read. And since it's the last line, we don't have to put the print statement on it unless you want. We can simp but we can simply do a plus b. So when I run this, uh, we have 10 15 29 and we get 30 24, which is what you exp