# Negative correlation

Negative correlation |
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**Negative correlation** is a statistical relationship between two variables where one variable increases while the other decreases, or vice versa. In management, this could mean that as one factor increases, the other factor decreases in performance or productivity. For example, if a business increases its advertising budget, then the resulting sales may decrease. This is an example of negative correlation.

## Example of negative correlation

- A negative correlation exists between smoking cigarettes and life expectancy. As the number of cigarettes smoked increases, the life expectancy decreases.
- There is a negative correlation between the number of hours a student spends studying and the amount of time they spend playing video games. As the number of hours studying increases, the amount of time spent playing video games decreases.
- Another example of a negative correlation is the relationship between exercise and weight gain. As the amount of exercise increases, the amount of weight gained decreases.

## Formula of negative correlation

The formula for negative correlation is:

$$\rho = -\frac{Cov(X,Y)}{\sigma_X \sigma_Y}$$

Where:

$$\rho$$ : Correlation coefficient

$$Cov(X,Y)$$ : Covariance between two variables X and Y

$$\sigma_X$$ : Standard deviation of X

$$\sigma_Y$$ : Standard deviation of Y

This formula measures the strength and direction of the linear relationship between two variables X and Y. If the correlation coefficient is negative, it indicates that as one variable increases, the other decreases. For example, if the correlation coefficient between two variables X and Y is -1, then as X increases, Y decreases by the same amount.

The formula is derived from the formula for the covariance between two variables X and Y, which is:

$$Cov(X,Y) = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{n - 1}$$

Where:

$$x_i$$ : The ith observation of X

$$y_i$$ : The ith observation of Y

$$\bar{x}$$ : The mean of X

$$\bar{y}$$ : The mean of Y

n : The number of observations

The covariance measures how two variables change together, and the correlation coefficient normalizes this measure by dividing it by the standard deviation of each variable. This allows us to compare the strength of the relationship between two variables, even if they have different units or scales.

## When to use negative correlation

Negative correlation is a valuable tool for managers to use in order to understand the relationship between different factors in their business environment. It can be used to identify potential problems and opportunities as well as to help make decisions about how to allocate resources. Negative correlations can be used in the following situations:

- To identify potential risks – Negative correlations can help managers identify potential risks by highlighting areas where one factor is increasing while another is decreasing. This can help managers make informed decisions about whether or not to invest in a particular area.
- To understand customer behavior – By studying the relationship between customer behavior and different factors, such as price, product features, and customer service, managers can gain insight into customer preferences and how to better meet their needs.
- To assess market trends – By looking at the relationship between different factors in the market, such as pricing, customer demand, and product features, managers can gain an understanding of the current market trends and make informed decisions about how to allocate resources.
- To optimize operations – By studying the relationship between different factors within a company’s operations, such as cost, time, and quality, managers can identify areas of inefficiency and take steps to optimize operations.

## Types of negative correlation

Negative correlation is a statistical relationship between two variables where one variable increases while the other decreases, or vice versa. In management, there are several types of negative correlation that can be observed. These include:

**Direct negative correlation**: This is a situation where a direct relationship exists between two variables, and when one increases, the other decreases. For example, an increase in employees may result in a decrease in profits.**Indirect negative correlation**: This is a situation where an indirect relationship exists between two variables, and when one increases, the other decreases. For example, an increase in gas prices may result in a decrease in sales of cars.**Inverse correlation**: This is a situation where the relationship between two variables is reversed, and when one increases, the other decreases. For example, an increase in the number of competitors may result in a decrease in profits.**Nonlinear correlation**: This is a situation where a nonlinear relationship exists between two variables, and when one increases, the other decreases. For example, an increase in the price of a product may result in a decrease in the number of sales.

## Limitations of negative correlation

Negative correlation has certain limitations that should be taken into consideration when using it to analyze data. These limitations include:

- It does not provide any information about the cause or effect of the relationship. Negative correlation only shows that two variables are related, but it does not explain why they are related.
- It may not be a reliable indicator of the strength of the relationship between two variables. A small difference in one variable may cause a large change in the other, which may not be accurately reflected in the correlation data.
- It may only be applicable to a certain range of values, and may not be applicable to all data sets.
- It may be influenced by outliers or extreme values, which can skew the results.
- It is only applicable to linear relationships, and may not be accurate for non-linear relationships.

Negative correlation can be explored through various approaches. These include:

**Observation and Monitoring**: Gathering data on the two variables over time to identify any changes in the relationship.**Correlation Analysis**: Looking for patterns in the data and using statistical tests to measure the strength of the correlation.**Causal Analysis**: Examining the cause and effect relationship between the variables.**Trend Analysis**: Examining the trend of the data over time to identify any changes in the relationship.

In conclusion, negative correlation can be explored through observation and monitoring, correlation analysis, causal analysis and trend analysis. These approaches can provide insights into the relationship between the two variables and help to identify any changes that may occur over time.

## Suggested literature

- Liu, Y., & Yao, X. (1999).
*Ensemble learning via negative correlation*. Neural networks, 12(10), 1399-1404. - Liu, Y., Yao, X., & Higuchi, T. (2000).
*Evolutionary ensembles with negative correlation learning*. IEEE Transactions on Evolutionary Computation, 4(4), 380-387. - Wang, S., Chen, H., & Yao, X. (2010, July).
*Negative correlation learning for classification ensembles*. In The 2010 international joint conference on neural networks (IJCNN) (pp. 1-8). IEEE.