In economics, calculus is used to study and record complex information — commonly on graphs and curves. Calculus allows for the determination of a maximal profit by providing an easy way to calculate marginal cost and marginal revenue. It can also be used to study supply and demand curves. Economists use assumptions in order to simplify economics processes so that they are easier to understand.
As a field, economics deals with complex processes and studies substantial amounts of information. Economists use assumptions in order to simplify economic processes so that it is easier to understand. Simplifying assumptions are used to gain a better understanding about economic issues with regards to the world and human behavior. Simple indifference curve : An indifference curve is used to show potential demand patterns.
It is an example of a graph that works with simplifying assumptions to gain a better understanding of the world and human behavior in relation to economics. Assumptions provide a way for economists to simplify economic processes and make them easier to study and understand. An assumption allows an economist to break down a complex process in order to develop a theory and realm of understanding.
Good simplification will allow the economists to focus only on the most relevant variables. Later, the theory can be applied to more complex scenarios for additional study. For example, economists assume that individuals are rational and maximize their utilities. This simplifying assumption allows economists to build a structure to understand how people make choices and use resources.
In reality, all people act differently. However, using the assumption that all people are rational enables economists study how people make choices.
Although, simplifying assumptions help economists study complex scenarios and events, there are criticisms to using them. Critics have stated that assumptions cause economists to rely on unrealistic, unverifiable, and highly simplified information that in some cases simplifies the proofs of desired conclusions. Examples of such assumptions include perfect information, profit maximization, and rational choices.
Economists use the simplified assumptions to understand complex events, but criticism increases when they base theories off the assumptions because assumptions do not always hold true. Although simplifying can lead to a better understanding of complex phenomena, critics explain that the simplified, unrealistic assumptions cannot be applied to complex, real world situations.
Economics, as a science, follows the scientific method in order to study data, observe patterns, and predict results of stimuli. There are specific steps that must be followed when using the scientific method. Economics follows these steps in order to study data and build principles:. Scientific Method : The scientific method is used in economics to study data, observe patterns, and predict results. Observation of data is critical for economists because they take the results and interpret them in a meaningful way.
Cause and effect relationships are used to establish economic theories and principles. Over time, if a theory or principle becomes accepted as universally true, it becomes a law. In general, a law is always considered to be true. The scientific method provides the framework necessary for the progression of economic study. All economic theories, principles, and laws are generalizations or abstractions. Through the use of the scientific method, economists are able to break down complex economic scenarios in order to gain a deeper understanding of critical data.
In economics, a model is defined as a theoretical construct that represents economic processes through a set of variables and a set of logical or quantitative relationships between the two. A model is simply a framework that is designed to show complex economic processes.
Most models use mathematical techniques in order to investigate, theorize, and fit theories into economic situations. Economists use models in order to study and portray situations. The focus of a model is to gain a better understanding of how things work, to observe patterns, and to predict the results of stimuli. Models are based on theory and follow the rules of deductive logic. Economic model diagram : In economics, models are used in order to study and portray situations and gain a better understand of how things work.
Economic models have two functions: 1 to simplify and abstract from observed data, and 2 to serve as a means of selection of data based on a paradigm of econometric study. Economic processes are known to be enormously complex, so simplification to gain a clearer understanding is critical. Selecting the correct data is also very important because the nature of the model will determine what economic facts are studied and how they will be compiled.
More generally, this knowledge makes it possible to find out what happens to the area of a square no matter how the length of its sides is changed, and conversely, how any change in the area affects the sides. Although they began in the concrete experience of counting and measuring, they have come through many layers of abstraction and now depend much more on internal logic than on mechanical demonstration. The test for the validity of new ideas is whether they are consistent and whether they relate logically to the other rules.
Mathematical processes can lead to a kind of model of a thing, from which insights can be gained about the thing itself. Any mathematical relationships arrived at by manipulating abstract statements may or may not convey something truthful about the thing being modeled.
However, if 2 cups of sugar are added to 3 cups of hot tea and the same operation is used, 5 is an incorrect answer, for such an addition actually results in only slightly more than 4 cups of very sweet tea. To be able to use and interpret mathematics well, therefore, it is necessary to be concerned with more than the mathematical validity of abstract operations and to also take into account how well they correspond to the properties of the things represented. Sometimes common sense is enough to enable one to decide whether the results of the mathematics are appropriate.
For example, to estimate the height 20 years from now of a girl who is 5' 5" tall and growing at the rate of an inch per year, common sense suggests rejecting the simple "rate times time" answer of 7' 1" as highly unlikely, and turning instead to some other mathematical model, such as curves that approach limiting values. Often a single round of mathematical reasoning does not produce satisfactory conclusions, and changes are tried in how the representation is made or in the operations themselves.
Indeed, jumps are commonly made back and forth between steps, and there are no rules that determine how to proceed. The process typically proceeds in fits and starts, with many wrong turns and dead ends. This process continues until the results are good enough. But what degree of accuracy is good enough? The answer depends on how the result will be used, on the consequences of error, and on the likely cost of modeling and computing a more accurate answer.
For example, an error of 1 percent in calculating the amount of sugar in a cake recipe could be unimportant, whereas a similar degree of error in computing the trajectory for a space probe could be disastrous.
The importance of the "good enough" question has led, however, to the development of mathematical processes for estimating how far off results might be and how much computation would be required to obtain the desired degree of accuracy.
This is so for several reasons, including the following: The alliance between science and mathematics has a long history, dating back many centuries. Science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analyzing data.
Often, abstract patterns that have been studied for their own sake by mathematicians have turned out much later to be very useful in science. Science and mathematics are both trying to discover general patterns and relationships, and in this sense they are part of the same endeavor. Mathematics is the chief language of science. The symbolic language of mathematics has turned out to be extremely valuable for expressing scientific ideas unambiguously.
Mathematics and science have many features in common. Clustering is used in unsupervised learning. Features may be represented as continuous, discrete, or discrete binary variables.
A feature is a function of one or more measurements, computed so that it quantifies some significant characteristics of the object. Example: consider our face then eyes, ears, nose, etc are features of the face. A set of features that are taken together, forms the features vector. Example: In the above example of a face, if all the features eyes, ears, nose, etc are taken together then the sequence is a feature vector [eyes, ears, nose].
The feature vector is the sequence of a feature represented as a d-dimensional column vector. The sequence of the first 13 features forms a feature vector. Pattern recognition possesses the following features: Pattern recognition system should recognize familiar patterns quickly and accurate Recognize and classify unfamiliar objects Accurately recognize shapes and objects from different angles Identify patterns and objects even when partly hidden Recognize patterns quickly with ease, and with automaticity.
Training and Learning in Pattern Recognition Learning is a phenomenon through which a system gets trained and becomes adaptable to give results in an accurate manner. Learning is the most important phase as to how well the system performs on the data provided to the system depends on which algorithms are used on the data. The entire dataset is divided into two categories, one which is used in training the model i. Training set, and the other that is used in testing the model after training, i.
Testing set. Training set: The training set is used to build a model. It consists of the set of images that are used to train the system. Training rules and algorithms are used to give relevant information on how to associate input data with output decisions.
The system is trained by applying these algorithms to the dataset, all the relevant information is extracted from the data, and results are obtained.
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