Executive Summary

ARIMA Model Overview and Key Metrics

MS

Model Summary

ARIMA Model Specification

ARIMA(0,1,1)
Model type

Model Summary model_summary ARIMA model specification and fit statistics

ARIMA(0,1,1)
model type
717
aic
728
bic
-354
log likelihood
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Key Insights

Model Summary

The provided ARIMA model summary indicates an ARIMA(0,1,1) model. Here’s what each part of the model specification means:

  1. ARIMA: Stands for Autoregressive Integrated Moving Average. It is a popular time series model used for forecasting.
  2. (0,1,1): The numbers in the parentheses represent the orders of the ARIMA model. In this case, the model has a differencing order of 1 (denoted by ‘1’ after the ‘1’ in parentheses), indicating that the raw data were differenced once to achieve stationarity. There is no autoregressive component (order ‘0’), but there is a moving average component of order 1 (order ‘1’).

The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are statistical measures used for model selection. Lower values of AIC and BIC indicate a better model fit relative to other candidate models.

In this particular case:

  • AIC = 716.96
  • BIC = 727.66
  • Log-likelihood = -354.48

The AIC and BIC values of this ARIMA(0,1,1) model are 716.96 and 727.66, respectively. These values are numerical indicators of the model’s goodness of fit. A lower AIC/BIC suggests a better trade-off between model fit and complexity.

The selection of this specific ARIMA(0,1,1) model may have been determined by a combination of factors such as the model’s fit statistics, diagnostic checks, and possibly the simplicity of the model. In this case, the lack of autoregressive terms and the inclusion of a moving average term with differencing may have been found to adequately capture the time series patterns while maintaining a parsimonious model structure.

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Key Insights

Model Summary

The provided ARIMA model summary indicates an ARIMA(0,1,1) model. Here’s what each part of the model specification means:

  1. ARIMA: Stands for Autoregressive Integrated Moving Average. It is a popular time series model used for forecasting.
  2. (0,1,1): The numbers in the parentheses represent the orders of the ARIMA model. In this case, the model has a differencing order of 1 (denoted by ‘1’ after the ‘1’ in parentheses), indicating that the raw data were differenced once to achieve stationarity. There is no autoregressive component (order ‘0’), but there is a moving average component of order 1 (order ‘1’).

The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are statistical measures used for model selection. Lower values of AIC and BIC indicate a better model fit relative to other candidate models.

In this particular case:

  • AIC = 716.96
  • BIC = 727.66
  • Log-likelihood = -354.48

The AIC and BIC values of this ARIMA(0,1,1) model are 716.96 and 727.66, respectively. These values are numerical indicators of the model’s goodness of fit. A lower AIC/BIC suggests a better trade-off between model fit and complexity.

The selection of this specific ARIMA(0,1,1) model may have been determined by a combination of factors such as the model’s fit statistics, diagnostic checks, and possibly the simplicity of the model. In this case, the lack of autoregressive terms and the inclusion of a moving average term with differencing may have been found to adequately capture the time series patterns while maintaining a parsimonious model structure.

FR

Forecast Results

Key Predictions

12
Periods ahead

Forecast Results forecast_results Future predictions with confidence intervals

12
periods ahead
95%
confidence level
508
mean forecast
479.97 - 552.06
forecast range
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Key Insights

Forecast Results

The forecast results provide valuable insights into future predictions with a 95% confidence level for a particular metric. In this case, the forecast is for a period 12 time units ahead, and the mean forecast value is 508.49. The forecast range is from 479.97 to 552.06, showing the potential highs and lows of the prediction within the confidence level.

The confidence level of 95% indicates that there is a high level of certainty in the forecast results, with only a 5% chance of the actual value falling outside the provided range. This level of confidence is commonly used in statistical analysis to balance accuracy and risk.

For business planning, these forecast values can be used as a guide for decision-making and resource allocation. The mean forecast of 508.49 serves as the central point around which plans can be made. The forecast range provides a sense of the possible variability and helps in understanding the level of uncertainty associated with the prediction.

By taking into account both the mean forecast and the forecast range, businesses can make informed decisions on aspects such as production levels, inventory management, budgeting, and overall strategy for the period ahead. It is essential to consider the confidence interval provided when interpreting the forecast results to ensure robust planning and risk management.

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Key Insights

Forecast Results

The forecast results provide valuable insights into future predictions with a 95% confidence level for a particular metric. In this case, the forecast is for a period 12 time units ahead, and the mean forecast value is 508.49. The forecast range is from 479.97 to 552.06, showing the potential highs and lows of the prediction within the confidence level.

The confidence level of 95% indicates that there is a high level of certainty in the forecast results, with only a 5% chance of the actual value falling outside the provided range. This level of confidence is commonly used in statistical analysis to balance accuracy and risk.

For business planning, these forecast values can be used as a guide for decision-making and resource allocation. The mean forecast of 508.49 serves as the central point around which plans can be made. The forecast range provides a sense of the possible variability and helps in understanding the level of uncertainty associated with the prediction.

By taking into account both the mean forecast and the forecast range, businesses can make informed decisions on aspects such as production levels, inventory management, budgeting, and overall strategy for the period ahead. It is essential to consider the confidence interval provided when interpreting the forecast results to ensure robust planning and risk management.

Time Series Forecast

Historical Data and Future Predictions

TS

Time Series Forecast

Historical and Predicted Values

Time Series Forecast — Historical data with forecasted values

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Key Insights

Time Series Forecast

The time series visualization labeled as “Time Series Forecast” presents historical data alongside forecasted values and confidence intervals. This type of visualization is commonly used to analyze trends, identify patterns, and predict future values based on historical data.

From the historical patterns observed in the visualization, it is possible to identify trends, seasonality, cyclical patterns, and any irregular fluctuations present in the data over time. The forecasted values extend from the historical data based on the identified patterns and any underlying relationships captured by the forecasting model utilized.

Concerning patterns or anomalies in the data may include sudden spikes or drops, persistent oscillations, outliers, or any inconsistencies that deviate significantly from the typical patterns in the time series. These anomalies could impact the accuracy of the forecasted values and may require further investigation to understand their cause and potential implications for future predictions.

To provide more detailed insights or detect specific anomalies, additional information such as the specific dataset, time period covered, forecasting method used, and any notable events influencing the data would be helpful.

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Key Insights

Time Series Forecast

The time series visualization labeled as “Time Series Forecast” presents historical data alongside forecasted values and confidence intervals. This type of visualization is commonly used to analyze trends, identify patterns, and predict future values based on historical data.

From the historical patterns observed in the visualization, it is possible to identify trends, seasonality, cyclical patterns, and any irregular fluctuations present in the data over time. The forecasted values extend from the historical data based on the identified patterns and any underlying relationships captured by the forecasting model utilized.

Concerning patterns or anomalies in the data may include sudden spikes or drops, persistent oscillations, outliers, or any inconsistencies that deviate significantly from the typical patterns in the time series. These anomalies could impact the accuracy of the forecasted values and may require further investigation to understand their cause and potential implications for future predictions.

To provide more detailed insights or detect specific anomalies, additional information such as the specific dataset, time period covered, forecasting method used, and any notable events influencing the data would be helpful.

Model Diagnostics

Decomposition and Correlation Analysis

DC

Time Series Decomposition

Trend, Seasonal, and Remainder Components

Decomposition — decomposition — Time series components breakdown

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Key Insights

Time Series Decomposition

From the provided data profile, it is clear that a time series has been decomposed into three main components: trend, seasonal, and remainder. Let’s discuss these components and their implications for the business:

  1. Trend: The trend component represents the long-term direction in which the time series is moving. A positive trend indicates a consistent increase over time, while a negative trend suggests a consistent decrease. Understanding the trend can help businesses in forecasting future performance and identifying overall growth or decline patterns. For example, a positive trend in sales could signal increasing demand and potential market expansion opportunities.

  2. Seasonal: The seasonal component captures repetitive patterns or fluctuations that occur within shorter time frames, such as monthly, quarterly, or annually. Seasonality is crucial for businesses to anticipate and plan for changes in demand throughout the year. For instance, retail businesses may experience higher sales during the holiday season, which can inform inventory management and marketing strategies.

  3. Remainder: The remainder component represents the residuals or random fluctuations that cannot be explained by the trend or seasonal patterns. It reflects unpredictable variations in the time series data. Understanding the remainder component helps businesses assess the accuracy of their forecasting models and identify any unexpected factors influencing the data. Monitoring the remainder can guide businesses in adjusting their strategies to address unforeseen challenges or opportunities.

In summary, analyzing the trend, seasonal, and remainder components of a time series can provide valuable insights for businesses in terms of forecasting future trends, adapting to seasonal fluctuations, and detecting unexpected variations in data. By leveraging this decomposition analysis, organizations can make informed decisions to optimize operations, improve planning efficiency, and enhance overall performance.

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Key Insights

Time Series Decomposition

From the provided data profile, it is clear that a time series has been decomposed into three main components: trend, seasonal, and remainder. Let’s discuss these components and their implications for the business:

  1. Trend: The trend component represents the long-term direction in which the time series is moving. A positive trend indicates a consistent increase over time, while a negative trend suggests a consistent decrease. Understanding the trend can help businesses in forecasting future performance and identifying overall growth or decline patterns. For example, a positive trend in sales could signal increasing demand and potential market expansion opportunities.

  2. Seasonal: The seasonal component captures repetitive patterns or fluctuations that occur within shorter time frames, such as monthly, quarterly, or annually. Seasonality is crucial for businesses to anticipate and plan for changes in demand throughout the year. For instance, retail businesses may experience higher sales during the holiday season, which can inform inventory management and marketing strategies.

  3. Remainder: The remainder component represents the residuals or random fluctuations that cannot be explained by the trend or seasonal patterns. It reflects unpredictable variations in the time series data. Understanding the remainder component helps businesses assess the accuracy of their forecasting models and identify any unexpected factors influencing the data. Monitoring the remainder can guide businesses in adjusting their strategies to address unforeseen challenges or opportunities.

In summary, analyzing the trend, seasonal, and remainder components of a time series can provide valuable insights for businesses in terms of forecasting future trends, adapting to seasonal fluctuations, and detecting unexpected variations in data. By leveraging this decomposition analysis, organizations can make informed decisions to optimize operations, improve planning efficiency, and enhance overall performance.

CA

Correlation Analysis

ACF and PACF

Correlation Analysis — Autocorrelation and partial autocorrelation functions

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Key Insights

Correlation Analysis

Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are crucial tools in time series analysis that help in understanding the patterns of correlation in the data.

When examining ACF patterns, the number of significant lags indicates the level of persistence in the data. For instance, if there is a significant lag at one period, it suggests the current value of the series is correlated with the previous value (lag 1 autocorrelation). If there are periodic significant lags at regular intervals, it implies a seasonality pattern in the data.

PACF, on the other hand, helps to identify the direct relationship between data points while controlling for the intervening values. Significant PACF at lag k indicates a direct correlation between data points k time units apart.

These autocorrelation patterns can inform model selection in a few ways:

  1. Identifying the order of autoregressive (AR) and moving average (MA) terms in ARIMA modeling based on the significant lags in ACF and PACF.
  2. Informing whether the series is stationary or exhibits seasonality, which can guide the selection of appropriate transformations or seasonal components in the model.
  3. Evaluating if the residual errors in a model contain any autocorrelation, which can guide adjustments to the model specification.

By analyzing ACF and PACF, analysts can gain insights into the underlying time series patterns and make informed decisions regarding model selection and forecasting.

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Key Insights

Correlation Analysis

Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are crucial tools in time series analysis that help in understanding the patterns of correlation in the data.

When examining ACF patterns, the number of significant lags indicates the level of persistence in the data. For instance, if there is a significant lag at one period, it suggests the current value of the series is correlated with the previous value (lag 1 autocorrelation). If there are periodic significant lags at regular intervals, it implies a seasonality pattern in the data.

PACF, on the other hand, helps to identify the direct relationship between data points while controlling for the intervening values. Significant PACF at lag k indicates a direct correlation between data points k time units apart.

These autocorrelation patterns can inform model selection in a few ways:

  1. Identifying the order of autoregressive (AR) and moving average (MA) terms in ARIMA modeling based on the significant lags in ACF and PACF.
  2. Informing whether the series is stationary or exhibits seasonality, which can guide the selection of appropriate transformations or seasonal components in the model.
  3. Evaluating if the residual errors in a model contain any autocorrelation, which can guide adjustments to the model specification.

By analyzing ACF and PACF, analysts can gain insights into the underlying time series patterns and make informed decisions regarding model selection and forecasting.

Statistical Analysis

Model Validation and Testing

ST

Stationarity Tests

ADF, KPSS, and Phillips-Perron

3

Stationarity Tests stationarity_tests Testing for time series stationarity

Test Statistic P_Value Result
Augmented Dickey-Fuller -3.292 0.076 Non-stationary
KPSS 2.478 0.010 Non-stationary
Phillips-Perron -21.804 0.041 Stationary
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Key Insights

Stationarity Tests

Based on the provided stationarity test results:

  1. Augmented Dickey-Fuller (ADF) Test:
    • Statistic: -3.2918
    • P-Value: 0.0759
    • Result: Non-stationary

The ADF test is a common test for stationarity. In this case, the test statistic is below the critical values, but the p-value is greater than 0.05. Since the p-value is higher than the significance level, we fail to reject the null hypothesis of non-stationarity.

  1. KPSS Test:
    • Statistic: 2.4784
    • P-Value: 0.01
    • Result: Non-stationary

The KPSS test is another stationarity test. Here, the test statistic exceeds the critical value, leading to rejection of the null hypothesis of stationarity in favor of non-stationarity.

  1. Phillips-Perron Test:
    • Statistic: -21.8041
    • P-Value: 0.0412
    • Result: Stationary

The Phillips-Perron test also assesses stationarity. In this instance, the test statistic is significant, and the p-value is below 0.05, indicating that the series is stationary.

Regarding stationarity and its importance for ARIMA models:

  • Stationarity refers to a time series where the mean, variance, and autocorrelation structure do not change over time. It is a crucial assumption for many time series models like ARIMA.
  • Stationarity is important for ARIMA models as these models assume a stationary series. Non-stationary data can lead to incorrect model predictions and unreliable parameter estimates.
  • If the data is found to be non-stationary through these tests, differencing may be needed to achieve stationarity by removing trends or seasonality from the series before modeling with ARIMA or other similar methods.

In summary, the result interpretation indicates that the time series may require differencing to achieve stationarity before being suitable for ARIMA modeling, as indicated by the ADF and KPSS tests. The Phillips-Perron test result suggests that the data may already be stationary based on that test.

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Key Insights

Stationarity Tests

Based on the provided stationarity test results:

  1. Augmented Dickey-Fuller (ADF) Test:
    • Statistic: -3.2918
    • P-Value: 0.0759
    • Result: Non-stationary

The ADF test is a common test for stationarity. In this case, the test statistic is below the critical values, but the p-value is greater than 0.05. Since the p-value is higher than the significance level, we fail to reject the null hypothesis of non-stationarity.

  1. KPSS Test:
    • Statistic: 2.4784
    • P-Value: 0.01
    • Result: Non-stationary

The KPSS test is another stationarity test. Here, the test statistic exceeds the critical value, leading to rejection of the null hypothesis of stationarity in favor of non-stationarity.

  1. Phillips-Perron Test:
    • Statistic: -21.8041
    • P-Value: 0.0412
    • Result: Stationary

The Phillips-Perron test also assesses stationarity. In this instance, the test statistic is significant, and the p-value is below 0.05, indicating that the series is stationary.

Regarding stationarity and its importance for ARIMA models:

  • Stationarity refers to a time series where the mean, variance, and autocorrelation structure do not change over time. It is a crucial assumption for many time series models like ARIMA.
  • Stationarity is important for ARIMA models as these models assume a stationary series. Non-stationary data can lead to incorrect model predictions and unreliable parameter estimates.
  • If the data is found to be non-stationary through these tests, differencing may be needed to achieve stationarity by removing trends or seasonality from the series before modeling with ARIMA or other similar methods.

In summary, the result interpretation indicates that the time series may require differencing to achieve stationarity before being suitable for ARIMA modeling, as indicated by the ADF and KPSS tests. The Phillips-Perron test result suggests that the data may already be stationary based on that test.

RD

Residual Diagnostics

Model Residual Analysis

18.034

Residual Diagnostics — Analysis of model residuals

18.034
ljung box statistic
0.585
ljung box pvalue
1.647
residuals mean
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Key Insights

Residual Diagnostics

Based on the provided data profile, the Ljung-Box test was conducted on the residuals of a model. The Ljung-Box test is used to assess whether autocorrelation is present in the residuals. The p-value of 0.5852 suggests that there is no strong evidence to reject the null hypothesis that the residuals are uncorrelated or that they exhibit white noise behavior.

In this case, since the p-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. This indicates that the residuals appear to be random and independent, showing characteristics of white noise behavior.

Additionally, knowing that the residuals have a mean of 1.6466 and a standard deviation of 5.8421 can provide further context for understanding the distribution and variability of the residuals. These metrics help to assess whether the residuals are centered around zero and how spread out they are.

Overall, based on the given information, the residuals show white noise behavior, suggesting that the model captures the underlying patterns well and that there is no significant autocorrelation present in the residuals.

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Key Insights

Residual Diagnostics

Based on the provided data profile, the Ljung-Box test was conducted on the residuals of a model. The Ljung-Box test is used to assess whether autocorrelation is present in the residuals. The p-value of 0.5852 suggests that there is no strong evidence to reject the null hypothesis that the residuals are uncorrelated or that they exhibit white noise behavior.

In this case, since the p-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. This indicates that the residuals appear to be random and independent, showing characteristics of white noise behavior.

Additionally, knowing that the residuals have a mean of 1.6466 and a standard deviation of 5.8421 can provide further context for understanding the distribution and variability of the residuals. These metrics help to assess whether the residuals are centered around zero and how spread out they are.

Overall, based on the given information, the residuals show white noise behavior, suggesting that the model captures the underlying patterns well and that there is no significant autocorrelation present in the residuals.

QQ

Normality Check

Q-Q Plot of Residuals

Normality Check — Q-Q plot to assess normality of residuals

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Key Insights

Normality Check

The Q-Q plot is a graphical method used to assess whether a set of data follows a specific distribution, such as the normal distribution. In this case, the Q-Q plot is used to check the normality of residuals from a model.

If the residuals in the Q-Q plot fall along a straight line, it indicates that the residuals are normally distributed. This implies that the assumptions of the statistical model are met, and the forecasts made using these residuals are reliable.

On the other hand, if the Q-Q plot shows deviations from a straight line, it suggests that the residuals may not be normally distributed. Non-normality of residuals can impact the validity of the model’s forecasts and confidence intervals. If the residuals are not normally distributed, forecast intervals may be biased, leading to inaccurate predictions and unreliable estimates of uncertainty.

Therefore, it is crucial to interpret the Q-Q plot results carefully to ensure the normality assumption holds for the residuals, thus ensuring the robustness and reliability of the forecasting model.

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Key Insights

Normality Check

The Q-Q plot is a graphical method used to assess whether a set of data follows a specific distribution, such as the normal distribution. In this case, the Q-Q plot is used to check the normality of residuals from a model.

If the residuals in the Q-Q plot fall along a straight line, it indicates that the residuals are normally distributed. This implies that the assumptions of the statistical model are met, and the forecasts made using these residuals are reliable.

On the other hand, if the Q-Q plot shows deviations from a straight line, it suggests that the residuals may not be normally distributed. Non-normality of residuals can impact the validity of the model’s forecasts and confidence intervals. If the residuals are not normally distributed, forecast intervals may be biased, leading to inaccurate predictions and unreliable estimates of uncertainty.

Therefore, it is crucial to interpret the Q-Q plot results carefully to ensure the normality assumption holds for the residuals, thus ensuring the robustness and reliability of the forecasting model.

Detailed Results

Coefficients and Forecast Values

CO

Model Coefficients

ARIMA Parameters

3

Model Coefficients model_coefficients ARIMA model parameter estimates

Parameter Estimate Std_Error
ma1 -0.704 0.069
sma1 -0.768 0.115
sma2 0.212 0.118
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Key Insights

Model Coefficients

The provided ARIMA model has 3 estimated parameters: MA(1), SMA(1), and SMA(2).

  1. Interpretation of Coefficients:

    • MA(1) Coefficient: The estimated coefficient for MA(1) is -0.7035. This indicates the impact of the lagged error term on the current value in the moving average component. A negative coefficient suggests that there is a negative linear relationship between the residuals of the model at lag 1 and the current observation.

    • SMA(1) and SMA(2) Coefficients: The estimated coefficients for SMA(1) and SMA(2) are -0.7679 and 0.2121, respectively. These coefficients represent the impact of the seasonal lagged error terms on the current value. SMA coefficients capture any remaining seasonality not accounted for by the AR and MA terms.

  2. Statistical Significance:

    • To evaluate the statistical significance of these coefficients, you typically compare the estimates to their standard errors. Lower standard errors relative to the estimate suggest more precision in estimation. In statistical analysis, coefficients are considered statistically significant if their p-values are less than a threshold (e.g., 0.05).
  3. Practical Importance:

    • In terms of practical importance, the coefficients indicate the strength and direction of the relationships between the lagged error terms and the current observation. It helps in understanding how past errors influence the present values and how seasonality impacts the series.

Please provide more details or let me know if you need further clarification on any specific aspect of the model coefficients.

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Key Insights

Model Coefficients

The provided ARIMA model has 3 estimated parameters: MA(1), SMA(1), and SMA(2).

  1. Interpretation of Coefficients:

    • MA(1) Coefficient: The estimated coefficient for MA(1) is -0.7035. This indicates the impact of the lagged error term on the current value in the moving average component. A negative coefficient suggests that there is a negative linear relationship between the residuals of the model at lag 1 and the current observation.

    • SMA(1) and SMA(2) Coefficients: The estimated coefficients for SMA(1) and SMA(2) are -0.7679 and 0.2121, respectively. These coefficients represent the impact of the seasonal lagged error terms on the current value. SMA coefficients capture any remaining seasonality not accounted for by the AR and MA terms.

  2. Statistical Significance:

    • To evaluate the statistical significance of these coefficients, you typically compare the estimates to their standard errors. Lower standard errors relative to the estimate suggest more precision in estimation. In statistical analysis, coefficients are considered statistically significant if their p-values are less than a threshold (e.g., 0.05).
  3. Practical Importance:

    • In terms of practical importance, the coefficients indicate the strength and direction of the relationships between the lagged error terms and the current observation. It helps in understanding how past errors influence the present values and how seasonality impacts the series.

Please provide more details or let me know if you need further clarification on any specific aspect of the model coefficients.

FT

Forecast Table

Detailed Predictions

12

Forecast Table forecast_table Detailed forecast values with confidence intervals

Period Forecast Lower_CI Upper_CI
1.000 492.700 479.970 505.430
2.000 499.240 485.960 512.520
3.000 504.110 490.310 517.920
4.000 506.210 491.900 520.520
5.000 506.180 491.380 520.980
6.000 504.230 488.960 519.510
7.000 498.310 482.580 514.040
8.000 505.190 489.010 521.370
9.000 508.170 491.560 524.780
10.000 521.510 504.480 538.550
11.000 521.810 504.360 539.260
12.000 534.210 516.360 552.060
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Key Insights

Forecast Table

Period-by-Period Forecast Interpretation:

  1. Period 1 (Forecast: 492.7):

    • The forecast for this period is 492.7, with lower and upper confidence intervals of 479.97 and 505.43, respectively.
  2. Period 2 (Forecast: 499.24):

    • The forecast for this period is 499.24, with confidence intervals ranging from 485.96 to 512.52.
  3. Period 3 (Forecast: 504.11):

    • The forecast for this period is 504.11, with a tighter confidence interval compared to the previous periods (490.31 to 517.92).
  4. Period 4 to Period 6:

    • The forecasts are relatively stable in this range, with slight variations in the upper and lower confidence intervals.
  5. Period 10 to Period 12:

    • In the later periods, the forecasts show an increasing trend, with wider confidence intervals reflecting higher uncertainty in the forecasted values.

Explanation of Confidence Interval Width Changes Over Time:

  • Narrower Intervals: Periods with narrow confidence intervals indicate higher certainty in the forecasted values, suggesting more confidence in the predicted outcomes during those periods.

  • Wider Intervals: Wider confidence intervals, especially in later periods, signify greater uncertainty in the forecasts. This widening of the intervals over time could be due to various factors such as increased volatility, data variability, or the inherent unpredictability of future trends.

  • Growing Confidence Uncertainty: The increasing width of confidence intervals towards later periods implies that as we forecast further into the future, the degree of uncertainty and potential variability in the predictions also grows. This may necessitate a closer monitoring and adjustment of forecasts as more data becomes available to refine the predictions.

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Key Insights

Forecast Table

Period-by-Period Forecast Interpretation:

  1. Period 1 (Forecast: 492.7):

    • The forecast for this period is 492.7, with lower and upper confidence intervals of 479.97 and 505.43, respectively.
  2. Period 2 (Forecast: 499.24):

    • The forecast for this period is 499.24, with confidence intervals ranging from 485.96 to 512.52.
  3. Period 3 (Forecast: 504.11):

    • The forecast for this period is 504.11, with a tighter confidence interval compared to the previous periods (490.31 to 517.92).
  4. Period 4 to Period 6:

    • The forecasts are relatively stable in this range, with slight variations in the upper and lower confidence intervals.
  5. Period 10 to Period 12:

    • In the later periods, the forecasts show an increasing trend, with wider confidence intervals reflecting higher uncertainty in the forecasted values.

Explanation of Confidence Interval Width Changes Over Time:

  • Narrower Intervals: Periods with narrow confidence intervals indicate higher certainty in the forecasted values, suggesting more confidence in the predicted outcomes during those periods.

  • Wider Intervals: Wider confidence intervals, especially in later periods, signify greater uncertainty in the forecasts. This widening of the intervals over time could be due to various factors such as increased volatility, data variability, or the inherent unpredictability of future trends.

  • Growing Confidence Uncertainty: The increasing width of confidence intervals towards later periods implies that as we forecast further into the future, the degree of uncertainty and potential variability in the predictions also grows. This may necessitate a closer monitoring and adjustment of forecasts as more data becomes available to refine the predictions.

AC

Accuracy Metrics

Model Performance

6.05
Rmse

Accuracy Metrics accuracy_metrics Model performance and error metrics

6.05
rmse
4.55
mae
1.9
mape
0.12
mase
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Key Insights

Accuracy Metrics

The accuracy metrics for the model are as follows:

  1. Root Mean Squared Error (RMSE): RMSE is a measure of the differences between predicted values by the model and the actual values observed. In this case, RMSE is 6.0462, indicating the average difference between the predicted and actual values is approximately 6.0462 units.

  2. Mean Absolute Error (MAE): MAE is the average of the absolute errors between predicted and actual values. A MAE of 4.5491 suggests that, on average, the model’s predictions are off by approximately 4.5491 units.

  3. Mean Absolute Percentage Error (MAPE): MAPE represents the average percentage difference between predicted and actual values. A MAPE value of 1.9% indicates that, on average, the model’s predictions have an error of 1.9% relative to the actual values.

  4. Mean Absolute Scaled Error (MASE): MASE compares the forecasted values from the model with the forecast of a naive method (e.g., simple moving average). A MASE value of 0.1196 suggests that the model’s forecasts are approximately 0.1196 times as accurate as the naive method.

In terms of forecast reliability, lower values of RMSE, MAE, MAPE, and MASE indicate better model performance. The closer these metrics are to zero, the more accurate the model is in predicting the target variable. A high RMSE, MAE, MAPE, or MASE value suggests that the model’s predictions deviate more from the actual values, indicating lower reliability in forecasting.

Therefore, based on the provided metrics, the model seems to have reasonable accuracy and reliability in its forecasts, as the values of RMSE, MAE, MAPE, and MASE are relatively low.

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Key Insights

Accuracy Metrics

The accuracy metrics for the model are as follows:

  1. Root Mean Squared Error (RMSE): RMSE is a measure of the differences between predicted values by the model and the actual values observed. In this case, RMSE is 6.0462, indicating the average difference between the predicted and actual values is approximately 6.0462 units.

  2. Mean Absolute Error (MAE): MAE is the average of the absolute errors between predicted and actual values. A MAE of 4.5491 suggests that, on average, the model’s predictions are off by approximately 4.5491 units.

  3. Mean Absolute Percentage Error (MAPE): MAPE represents the average percentage difference between predicted and actual values. A MAPE value of 1.9% indicates that, on average, the model’s predictions have an error of 1.9% relative to the actual values.

  4. Mean Absolute Scaled Error (MASE): MASE compares the forecasted values from the model with the forecast of a naive method (e.g., simple moving average). A MASE value of 0.1196 suggests that the model’s forecasts are approximately 0.1196 times as accurate as the naive method.

In terms of forecast reliability, lower values of RMSE, MAE, MAPE, and MASE indicate better model performance. The closer these metrics are to zero, the more accurate the model is in predicting the target variable. A high RMSE, MAE, MAPE, or MASE value suggests that the model’s predictions deviate more from the actual values, indicating lower reliability in forecasting.

Therefore, based on the provided metrics, the model seems to have reasonable accuracy and reliability in its forecasts, as the values of RMSE, MAE, MAPE, and MASE are relatively low.

Business Insights

Key Takeaways and Recommendations

BI

Business Insights

Key Recommendations

Increasing
Trend direction

Business Insights business_insights Actionable insights and recommendations

Increasing
trend direction
3.77
avg growth rate
30.84
uncertainty range
IN

Key Insights

Business Insights

Based on the ARIMA model forecast:

  1. Trend Direction: The trend is increasing at an average rate of 3.77 per period. This indicates positive growth in the value being forecasted.

  2. Forecasted Average Value: The ARIMA model projects an average value of 508.49 over the next 12 periods. This provides an estimate of the expected performance in the upcoming periods.

  3. Uncertainty Range: The uncertainty range of 30.84 suggests that there is variability around the average forecasted value. It is important to consider this range when making decisions based on the forecast.

Actionable Recommendations:

  1. Inventory Planning: With the forecasted increasing trend, it is advisable to ensure that inventory levels are adjusted to meet the expected demand. Consider increasing stock levels gradually to align with the projected growth rate.

  2. Staffing Recommendations: As the trend is upward, it might be beneficial to evaluate staffing needs to support the anticipated increase in business activity. Evaluate hiring or scheduling additional staff based on the growth rate to handle potential demand.

  3. Budget Planning: Given the uncertainty range, it is crucial to have a flexible budget that accounts for variations in the forecasted values. Allocate resources keeping in mind the range and potential growth to mitigate risks associated with fluctuations.

  4. Monitoring and Adjustments: Continuously monitor the actual performance against the forecast and make necessary adjustments to optimize operations. Regularly reassess the forecast to align strategies with real-time data and market conditions.

By incorporating these recommendations into your business planning, you can leverage the ARIMA model forecast to make informed decisions regarding inventory, staffing, and budget adjustments to capitalize on the projected growth and manage uncertainties effectively.

IN

Key Insights

Business Insights

Based on the ARIMA model forecast:

  1. Trend Direction: The trend is increasing at an average rate of 3.77 per period. This indicates positive growth in the value being forecasted.

  2. Forecasted Average Value: The ARIMA model projects an average value of 508.49 over the next 12 periods. This provides an estimate of the expected performance in the upcoming periods.

  3. Uncertainty Range: The uncertainty range of 30.84 suggests that there is variability around the average forecasted value. It is important to consider this range when making decisions based on the forecast.

Actionable Recommendations:

  1. Inventory Planning: With the forecasted increasing trend, it is advisable to ensure that inventory levels are adjusted to meet the expected demand. Consider increasing stock levels gradually to align with the projected growth rate.

  2. Staffing Recommendations: As the trend is upward, it might be beneficial to evaluate staffing needs to support the anticipated increase in business activity. Evaluate hiring or scheduling additional staff based on the growth rate to handle potential demand.

  3. Budget Planning: Given the uncertainty range, it is crucial to have a flexible budget that accounts for variations in the forecasted values. Allocate resources keeping in mind the range and potential growth to mitigate risks associated with fluctuations.

  4. Monitoring and Adjustments: Continuously monitor the actual performance against the forecast and make necessary adjustments to optimize operations. Regularly reassess the forecast to align strategies with real-time data and market conditions.

By incorporating these recommendations into your business planning, you can leverage the ARIMA model forecast to make informed decisions regarding inventory, staffing, and budget adjustments to capitalize on the projected growth and manage uncertainties effectively.