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Both Correlation and Covariance measures the linear association between two variables. To be specific and make it apparent, let us understand the key difference, Correlation measures the strength of a relationship between two variables. Covariance measures the direction of a relationship between two variables. Specific comparison: Values: Correlation: Standardized Covariance: Unstandardized Units: Correlation: Has Units Covariance: Does not have units Scale: Correlation: Change in scale does not affect the value of Correlation Covariance: Change in Scale will affect the value of Covariance Range: Correlation: -1 to +1 Covariance: -∞ to +∞ Why Correlation value lies between -1 and +1? Correlation is nothing but Covariance divided by standard deviation of the variables, hence the value lies between -1 and +1. Which means, it is scaled down version of covariance. Inferences from Analysis: Covariance Inference: Positive - Both the variables increase or decreases together - Directly Proportional Negative - Inverse, if one variable increases, the other decreases - Inversely Proportional Correlation Inference: +1 - Perfect Positive linear relationship 0 - No linear relationship -1 - Perfect Negative Linear relationship Some more examples: Correlation Examples: Pearson r Relationship 0 No relationship 0.466 Moderate positive relationship 0.95 Large positive relationship -0.96 Large negative relationship Covariance Examples: Covariance Relationship 0.0036 Positive 0 No variance -0.007 Negative -0.0376 Negative Covariance, typically can take any value and it is toilsome to interpret the number. Sample Data Set: G Price CO Price 49000 95.17 48600 98.4 48600 98.4 48600 98.4 48250 97.17 48000 97.16 47800 101.24 47800 101.24 47800 101.24 47950 103.66 Based on the same data set, below is association summary: Correlation(R) -0.74682 Covariance(G,CO) -744.37 There are numerous applications of Correlation and Covariance, some are listed below: Data science: one of the frequent used measurement is Covariance. Insights from covariance analysis can help us to get more clarity on Multivariate data. Stock market: Investors, traders and analyst often use correlation and covariance. Specifically, to understand the hidden correlation on the stock returns of one company to other, which could potentially bring down and minimize the investment risks. Implied Correlation Index by CBOE (Chicago Board Options Exchange): This tracks the correlation between implied volatilities of options and weighted portfolio of options Banking and Insurance: Exploratory analysis can give more insights on the variable relationship which assists in customer churn and retention.

1. Correlation depicts magnitude & direction of linear relation between two data series. The value of correlation for sample universe varies between range +1 to -1, depicting positive and negative slope for the regression line. Here, negative correlation values between -1 to 0 represent a negative association-- with magnitude of one data series value decreasing, as the magnitude of the other data series variable increases. Likewise, positive correlation values between 0 to +1 represent positive association-- with magnitude of one data series value increases, as the magnitude of the other data series value increases. Specifically, correlation values greater than -0.7 to -1, indicating strong negative correlation, values between -0.7 and -0.5, indicates modest negative correlation and -0.4 to 0, indicates weak or poor correlation. Similarly, correlation values greater than 0.7 to +1 indicating strong correlation, values between 0.7 and 0.5 indicates modest correlation and 0.4 to 0, indicates weak or poor correlation. 2. Schematically correlation may be represented as downward sloping line in scatter-plot diagram; however, the actual direction of correlation curve, would depend upon value of the data points. 3. Covariance represents the differences in position of data points of series from mean value. An upward facing line-plot may be used in scatter-plot diagram, to schematically represent deviation of data variables from mean value. The theoretical value of covariance between variable of a data series may range between -∞ to +∞. 4. Correlation phenomenon is observed between values of different data types, represented in different data series; whereas, the covariance phenomenon is observed between data values of similar data types represented in same series. Both Covariance & Correlation values, are used interchangeably (basis similarity/difference in variable values type & scale units), to estimate value of Eigen value and Eigen Vector, that is employed for estimation of the PCA principal component analysis. It is theoretically possible to have different estimate of PCA principal component analysis using covariance matrix or correlation matrix while reaching estimate of Eigen value and Eigen Vector. 5. Correlation value may be obtained by dividing the value of covariance, with the standard deviation of independent data series. 6. Correlation may be impacted by instance of autocorrelation between variable i.e., the explanation of dependent variable is not fully explained by independent predictor variable. This can be illustrated with help of residual plot, where residual variable, (calculated as difference between observation variable i.e., dependent variable plotted on y-axis & fitted value i.e., independent variable plotted on x-axis) exhibit correlation between two consecutive values. Hence, in case of auto correlation, error term (defined as difference between expected value and actual value) may explain/predict/infer values of dependent variable, instead of analyst expectation of value of dependent regressand variable explained by explanatory regressor covariate.

Santosh Mane has provided the best answer to this question. He has also provided an example to support his answer. Answer from Johanan is a must read to get a holistic understanding of P-Diagram.

In Six Sigma, the process is measured in terms of defects; it aims to develop a process that delivers only 3.4 defects per million opportunities (DPMO). Processes are defined as a combination of inputs, actions, and outputs, or as a series of ongoing activities that transform inputs into outputs for the customer. In Six Sigma strategy, Design for Six Sigma (DFSS) is about developing products that consistently do things right. In regular Six Sigma, DMAIC (Define- Measure- Analyze- Improve- Control) is used to improve a current process without changing the fundamental structure of the process. In a company, it is used when the existing product does not meet customer specifications or is not performing adequately. The DFSS method, on the other hand, is used to create optimized designs from the beginning of the process life cycle. Decisions made during the development stage have a significant impact on the final product performance, which is why DFSS has a close relationship with the product development process. As a result, the DFSS method shifts the focus from improving performance in the later phases of the design cycle (DMAIC) to the front-end phases. Let me now describe the Taguchi method. The Taguchi method, also called Robust Design Method, was developed by Dr. Genichi Taguchi to improve productivity. Product processes begin with inputs (such as user intent, energy, or other factors) and end with outputs that are functions delivered to the customer. This is the reason why the Taguchi method frequently uses a process diagram called the P-diagram to illustrate the process model for creating a product or manufacturing process. It consciously takes into account "noise factors" such as environmental variations during product usage, manufacturing variations, and component deterioration. Taguchi method also ensures customer satisfaction, improves the fundamental function of the product and thus, facilitates flexible designs and concurrent engineering. P-diagram is a tool used to classify the variables associated with the product into inputs, noise factors, error states, design control parameters and output/ideal response. In defining the development scope, the input (energy, signal, user intent, etc.) and output (product characteristics, functions, performance) associated with the design concept are first identified. After which, consideration is given to the factors beyond the control of the designer; these are called the noise factors. Design parameters or control factors which are the parameters that can be specified by the designer are then determined. These control factors will influence the output and both can be adjusted and controlled. However, noise factors will also influence output and cannot be significantly controlled; this causes the relationship to deviate from the idea. Thus, it is important to select appropriate control factors (design parameters) that will reduce the deviation from idea to minimum; such is called robust design. Let us see an example of implementing the CAPA process

Parameter Diagram In its simplicity, the Parameter Diagram can be considered a schematic representation of the signal, noise, control factors, and response variable. Juran defines Parameter Diagrams in his book on Quality, viz. Quality Planning and Analysis. He states that performance is one of the most basic features of any product and equates performance to the output of a product/service/system. For example, in a TV, it could be colour density, in an electric car it could be range, etc. The output is a factor of various engineering principles that are used to combine materials, components, parts, assemblies, etc. Each of these inputs has a range of parameters, which the design team needs to establish. Setting the correct value for each parameter would then produce the desired output. The Parameter Diagram relates the inputs which may come from the system or customer to the desired output. The parameter diagram also takes into consideration external influences that are non-controllable. The Parameter Diagram can be used as a visual tool for brainstorming and documenting various factors such as control factors, noise factors, input signals, error states, and ideal responses. Use Cases of P-Diagram - It can be used as a very effective Brainstorming Tool. - It is helpful in the preparation of an FMEA, as each input could be analyzed for the failure modes and the effects can be analyzed and RPN Number calculated. - In complex systems where there is an interaction between numerous subsystems, design and engineering parameters, and operating conditions, the Parameter diagram helps the team to visually see each element. - It can also be used for Component FMEA when the visual representation of inputs, ideal response, noise factors, and control factors is helpful. - It can also be used in New Product Development, in combination with the Design of Experiments. An Indicative Diagram of a Parameter Diagram is below:- Definitions of Elements of P Diagram - Input Signals. These are generally the energy sources required by the system. Variations in these inputs can cause a change in the output. - Control Factors. These are the parameters that the design team can control or change. - Error States. These are the loss of energy or undesirable output - Noise Factors. These are factors that influence the output but are not under the direct control of the design team. These include fair wear and tear of the equipment, normal degradation of materials, tiredness of operator, etc. A robust design protects the system against such noise factors. - Ideal Response. This is the desired functional output of the system. Parameter Diagram, New Product Development, and Design of Experiments DMADV. New Product Development generally follows the DMADV (Define, Measure, Analise, Design, Verify) methodology. In the Define phase, the process and its design goals are defined. In the Measure phase, the CTQ is identified and measured. In the Analyse phase data is used to analyze the best possible design. In the Design Phase, the prototype is designed and finally, in the Verify phase, the outputs are verified in real conditions. The DMADV methodology is used in New Product Development or when further improvement of the existing product is not a commercially or technologically viable solution. In the first case, no data would be available for analysis, and in the second case, the data from the previous product could possibly be used for analysis. Design of Experiments. Experimental design is a critically important tool in scientific and engineering for improving the product realization process. Critical components of these activities are new manufacturing process design and development, and process management. The application in DoE helps in increasing process yield, reduced variability, development time, and cost. Applications of DoE include - Evaluation and comparison of basic design configurations - Evaluation of material alternatives - Selection of design parameters so that the product will work under a wide variety of field conditions, that is so that the product is robust - Determination of key product design parameters that impact product performance - Formulation of New Products Parameter Diagram. After representing the system schematically in a Parameter Diagram, the Design of Experiments can be used with the input signals, control factors and error states, ideal response can be optimized based on the inputs. References https://www.weibull.com/hotwire/issue182/fmeacorner182.htm https://kanbanize.com/lean-management/six-sigma/dmadv Design and Analysis of Experiments, Eight Edition, Douglas C. Montgomery, Wiley